Quadratic equations
61 flashcards covering Quadratic equations for the ACT Math section.
Quadratic equations are mathematical expressions that involve a variable raised to the second power, typically written as ax² + bx + c = 0, where a, b, and c are constants and a isn't zero. They represent parabolic curves on a graph and are used to model real-world scenarios, like the path of a projectile or optimizing quantities in business. Understanding quadratics helps build a foundation for algebra and problem-solving, making them a key topic in math education.
On the ACT Math section, quadratic equations show up in multiple-choice questions that test solving for roots, factoring, completing the square, or using the quadratic formula. Common traps include errors in factoring, overlooking the discriminant (which indicates the number of real solutions), or misinterpreting word problems that disguise quadratics. Focus on practicing graph interpretation, identifying the vertex and intercepts, and applying the formula efficiently to handle time-pressured questions.
Remember to always check your solutions by substituting them back into the original equation.
Terms (61)
- 01
Quadratic equation
A polynomial equation of degree two, generally written as ax^2 + bx + c = 0, where a, b, and c are constants and a is not zero.
- 02
Standard form of a quadratic
The form ax^2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0, used as the basic structure for solving quadratic equations.
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Vertex form of a quadratic
The form y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola, helpful for graphing and identifying the maximum or minimum point.
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Factored form of a quadratic
The form a(x - r)(x - s) = 0, where r and s are the roots, allowing quick identification of solutions by setting each factor to zero.
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Quadratic formula
The equation x = [-b ± √(b^2 - 4ac)] / (2a) used to find the roots of ax^2 + bx + c = 0, applicable when factoring is difficult.
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Discriminant
The value b^2 - 4ac in a quadratic equation ax^2 + bx + c = 0, which determines the number of real roots: positive for two, zero for one, and negative for none.
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Roots of a quadratic equation
The values of x that satisfy ax^2 + bx + c = 0, representing the points where the parabola intersects the x-axis.
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Axis of symmetry
The vertical line x = -b/(2a) that divides a parabola into two mirror-image halves, useful for locating the vertex.
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Vertex of a parabola
The highest or lowest point on a parabola, given by the point (-b/(2a), f(-b/(2a))), indicating the maximum or minimum value.
- 10
Parabola
The U-shaped graph of a quadratic function y = ax^2 + bx + c, which opens upward if a > 0 and downward if a < 0.
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Factoring quadratics
A method to solve ax^2 + bx + c = 0 by rewriting it as a(x + p)(x + q) = 0, where p and q are numbers that multiply to c/a and add to b/a.
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Completing the square
A technique to rewrite a quadratic as (x + h)^2 + k = 0 by adding and subtracting a constant, useful for solving or converting to vertex form.
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Graphing a quadratic function
Plotting y = ax^2 + bx + c by finding the vertex, axis of symmetry, and intercepts, to visualize the equation's behavior.
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Maximum value of a quadratic
The y-coordinate of the vertex when a < 0, representing the highest point on a downward-opening parabola.
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Minimum value of a quadratic
The y-coordinate of the vertex when a > 0, representing the lowest point on an upward-opening parabola.
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Domain of a quadratic function
The set of all real numbers, as quadratic functions are defined for every x-value without restrictions.
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Range of a quadratic function
The set of y-values the function can take, starting from the vertex's y-coordinate and extending infinitely in one direction based on the parabola's orientation.
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Quadratic inequality
An inequality involving a quadratic expression, such as ax^2 + bx + c > 0, solved by finding roots and testing intervals.
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Solving quadratic inequalities
Determining the intervals where the quadratic expression is positive or negative by graphing or testing points around the roots.
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Systems of equations with quadratics
Solving a quadratic equation alongside a linear one, often by substitution or graphing, to find intersection points.
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Word problems involving quadratics
Real-world scenarios modeled by quadratic equations, such as projectile motion or area optimization, requiring setup and solving.
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Difference of squares
A factoring pattern for expressions like a^2 - b^2, which factors to (a - b)(a + b), useful for simplifying quadratics.
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Perfect square trinomial
A quadratic like a^2 + 2ab + b^2, which factors to (a + b)^2, helping in completing the square or solving.
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Sum of roots
For a quadratic ax^2 + bx + c = 0, the sum of the roots is -b/a, a property derived from the equation's coefficients.
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Product of roots
For a quadratic ax^2 + bx + c = 0, the product of the roots is c/a, useful for verifying solutions or forming equations.
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Vertex formula
The method to find the vertex coordinates as x = -b/(2a) and y = c - b^2/(4a), for quick graphing without full solving.
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X-intercept of a quadratic
The points where the parabola crosses the x-axis, found by solving ax^2 + bx + c = 0 for y = 0.
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Y-intercept of a quadratic
The point where the graph crosses the y-axis, which is the value of c when x = 0 in y = ax^2 + bx + c.
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Symmetry in parabolas
The property that points on the parabola are mirror images across the axis of symmetry, aiding in sketching graphs.
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Upward-opening parabola
A parabola that opens upward when the leading coefficient a is positive, resulting in a minimum point at the vertex.
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Downward-opening parabola
A parabola that opens downward when the leading coefficient a is negative, resulting in a maximum point at the vertex.
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Leading coefficient
The coefficient a in ax^2 + bx + c, which affects the width and direction of the parabola.
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Zero product property
The rule that if a product of factors equals zero, then at least one factor must be zero, fundamental for solving factored quadratics.
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Example: Factoring x^2 + 5x + 6 = 0
This equation factors to (x + 2)(x + 3) = 0, so the roots are x = -2 and x = -3.
Identify factors of 6 that add to 5: 2 and 3.
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Example: Completing the square for x^2 + 6x + 5 = 0
Rewrite as (x + 3)^2 - 4 = 0, so x + 3 = ±2, yielding roots x = -1 and x = -5.
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Example: Using quadratic formula for 2x^2 + 3x - 2 = 0
Apply x = [-3 ± √(9 + 16)] / 4, resulting in x = 0.5 and x = -2.
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Example: Finding discriminant of x^2 - 4x + 4 = 0
The discriminant is 16 - 16 = 0, indicating one real root.
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Example: Graphing y = x^2 - 4
The vertex is at (0, -4), it opens upward, and x-intercepts are at x = ±2.
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Common trap: Forgetting to check extraneous solutions
In equations derived from quadratics, solutions might not satisfy the original equation, so always verify by substitution.
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Common trap: Incorrectly factoring quadratics
Mistakes occur if factors don't multiply to the constant term and add to the middle coefficient, leading to wrong roots.
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Strategy: Check for factorability first
Before using the quadratic formula, try factoring if the discriminant is a perfect square and coefficients are simple.
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Strategy: Use quadratic formula for non-factorable equations
Apply the formula when coefficients are messy or factoring fails, ensuring accurate roots.
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Monomial in quadratics
A single-term expression like 3x^2, which is the basic building block but not a full quadratic unless combined.
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Binomial in quadratics
A two-term expression like x^2 + 1, often part of factoring or simplifying quadratic equations.
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Trinomial in quadratics
A three-term expression like x^2 + 5x + 6, which is the standard form of most quadratic equations.
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Degree of a polynomial
The highest power of x in a polynomial, which is 2 for quadratic equations, distinguishing them from linear or cubic ones.
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Constant term in a quadratic
The term c in ax^2 + bx + c, representing the y-intercept and affecting the roots' product.
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Rational roots
Roots that are fractions or integers, possible when the discriminant is a perfect square and coefficients are rational.
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Irrational roots
Roots that involve square roots, occurring when the discriminant is positive but not a perfect square.
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Complex roots
Roots with imaginary parts, arising when the discriminant is negative, though ACT focuses on real numbers.
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Number of real solutions
Determined by the discriminant: two if positive, one if zero, and none if negative for real numbers.
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Transformations of quadratics
Changes like shifting or stretching y = ax^2 + bx + c, such as y = a(x - h)^2 + k, to alter the graph's position and shape.
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Quadratic in vertex form applications
Using y = a(x - h)^2 + k to quickly find the vertex and sketch, especially in optimization problems.
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Area problems with quadratics
Equations modeling maximum area, like for a rectangle with fixed perimeter, solved to find optimal dimensions.
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Distance problems with quadratics
Scenarios like time to reach a point in motion, where distance formulas lead to quadratic equations.
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Profit maximization
Using quadratics to model revenue and cost, finding the vertex to determine maximum profit.
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Example: Solve inequality x^2 < 4
The roots are x = ±2, so the solution is -2 < x < 2, as the parabola is below the x-axis in that interval.
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Example: Find roots of 3x^2 - 2x - 1 = 0
Using the quadratic formula, x = [2 ± √(4 + 12)] / 6, yielding x = 1 and x = -1/3.
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Parabola intersection with a line
Solving a system like y = x^2 + 1 and y = 2x - 1 to find intersection points.
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Synthetic division for quadratics
A shortcut to divide a quadratic by a linear factor, though less common, to check roots.
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Quadratic regression
Fitting a quadratic equation to data points using technology, to model trends on the ACT.