Quadratic equations
58 flashcards covering Quadratic equations for the SAT Math section.
Quadratic equations are equations that involve a variable raised to the second power, typically written in the form ax² + bx + c = 0, where a, b, and c are constants and a isn't zero. They represent relationships that form a parabola when graphed, often used to model real-world scenarios like the trajectory of a projectile or finding the maximum height of an object. Understanding quadratics helps build a foundation in algebra for solving problems with two possible solutions.
On the SAT Math section, quadratic equations show up in multiple-choice and grid-in questions, testing skills like factoring, using the quadratic formula, or completing the square. Common traps include mistakes in factoring due to sign errors or overlooking extraneous solutions, as well as misinterpreting word problems that hide a quadratic relationship. Focus on recognizing when to apply the discriminant to determine the number of real roots and practicing graphing to visualize solutions quickly.
Double-check your answers by substituting them back into the original equation.
Terms (58)
- 01
Quadratic equation
An equation that can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants and a is not zero.
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Standard form of a quadratic
The form ax^2 + bx + c = 0, which is used to identify coefficients and apply solving methods.
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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, useful for graphing and finding maximum or minimum values.
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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 factors to zero.
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Roots of a quadratic equation
The values of x that satisfy ax^2 + bx + c = 0, representing where the parabola intersects the x-axis.
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Zero of a quadratic function
A root of the equation f(x) = ax^2 + bx + c = 0, indicating an x-value where the function equals zero.
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Discriminant
The expression b^2 - 4ac in a quadratic equation ax^2 + bx + c = 0, which determines the nature of the roots: positive for two real, zero for one real, negative for no real.
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Quadratic formula
The formula x = [-b ± sqrt(b^2 - 4ac)] / (2a) used to find the roots of ax^2 + bx + c = 0 when factoring is not straightforward.
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Factoring quadratics
A method to solve ax^2 + bx + c = 0 by rewriting it as a product of binomials, such as (x + p)(x + q) = 0, and setting each factor to zero.
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Completing the square
A technique to solve or rewrite a quadratic equation by turning ax^2 + bx + c into a perfect square trinomial, like a(x + d)^2 + e = 0.
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Graph of a parabola
The U-shaped curve produced by plotting y = ax^2 + bx + c, which opens upward if a > 0 and downward if a < 0.
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Vertex of a parabola
The highest or lowest point on the graph of y = ax^2 + bx + c, occurring at x = -b/(2a) and representing the maximum or minimum value.
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Axis of symmetry
The vertical line that passes through the vertex of a parabola, given by x = -b/(2a) for y = ax^2 + bx + c.
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Direction of a parabola
Determined by the coefficient a in y = ax^2 + bx + c; if a is positive, it opens upward; if negative, it opens downward.
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Maximum or minimum value
The y-coordinate of the vertex; for y = ax^2 + bx + c, it is the highest point if a < 0 or lowest if a > 0.
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Domain of a quadratic function
The set of all real numbers for which the function y = ax^2 + bx + c is defined, which is all x values.
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Range of a quadratic function
The set of y-values the function y = ax^2 + bx + c can take, starting from the vertex's y-value and extending infinitely in one direction.
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Solving quadratic inequalities
Finding the values of x that satisfy an inequality like ax^2 + bx + c > 0 by determining the roots and testing intervals on the number line.
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Sum of roots
For a quadratic ax^2 + bx + c = 0, the sum of the roots is -b/a, which can be used to verify solutions without fully solving.
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Product of roots
For a quadratic ax^2 + bx + c = 0, the product of the roots is c/a, helpful for checking factored forms or understanding root relationships.
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Difference of squares
A factoring pattern for expressions like a^2 - b^2, which equals (a - b)(a + b), often used for quadratics like x^2 - 4.
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Perfect square trinomial
An expression like (x + a)^2 = x^2 + 2ax + a^2, which is a quadratic that factors into identical binomials.
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Factoring by grouping
A method to factor quadratics or polynomials by grouping terms, such as factoring x^3 + 2x^2 + x + 2 into (x^3 + 2x^2) + (x + 2).
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Quadratic word problems
Problems that translate real-world scenarios into quadratic equations, such as finding the time an object is in the air using height formulas.
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Projectile motion equation
A quadratic equation like h = -16t^2 + vt + h0 for an object under gravity, used to find time of flight or maximum height.
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Optimization problems
Problems where a quadratic represents a quantity to maximize or minimize, like finding the dimensions of a rectangle with maximum area.
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Break-even analysis
Using a quadratic equation to find the point where costs equal revenue, often solving for the number of units that result in zero profit.
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Extraneous roots
Solutions that arise from manipulating equations but do not satisfy the original, such as in quadratics solved by squaring both sides.
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Graphing quadratics
Plotting points or using the vertex and intercepts to sketch y = ax^2 + bx + c, helping visualize roots and vertex.
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Y-intercept of a quadratic
The point where the graph crosses the y-axis, which is the value of c in y = ax^2 + bx + c.
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X-intercepts of a quadratic
The points where the graph crosses the x-axis, corresponding to the roots of ax^2 + bx + c = 0.
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Symmetry in quadratics
The property that the graph of y = ax^2 + bx + c is symmetric about its axis of symmetry, meaning points equidistant from the axis have the same y-value.
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Converting to vertex form
Rewriting ax^2 + bx + c as a(x - h)^2 + k by completing the square, which makes it easier to identify the vertex.
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Using discriminant for roots
Calculating b^2 - 4ac to predict if a quadratic has two distinct real roots, one real root, or no real roots.
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Rational roots
Roots of a quadratic that are rational numbers, often found when the equation factors nicely over integers.
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Irrational roots
Roots that are not rational, such as those involving square roots, as in the solutions to x^2 - 2 = 0.
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Complex roots
Roots that are not real numbers, occurring when the discriminant is negative, like in x^2 + 1 = 0.
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Strategy for choosing solving method
Deciding between factoring, completing the square, or quadratic formula based on the coefficients; for example, use factoring if the quadratic factors easily.
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Common traps in quadratics
Errors like forgetting to divide all terms by a leading coefficient or misinterpreting the discriminant as the roots themselves.
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Example: Factoring x^2 + 5x + 6
Factoring gives (x + 2)(x + 3) = 0, so roots are x = -2 and x = -3.
The equation x^2 + 5x + 6 = 0 factors to show solutions directly.
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Example: Completing the square for x^2 + 4x - 5
Rewrite as (x + 2)^2 - 9 = 0, so x + 2 = ±3, yielding x = -2 + 3 or x = -2 - 3.
Solutions are x = 1 and x = -5.
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Example: Quadratic formula for 2x^2 - 3x - 2
Apply x = [3 ± sqrt(9 + 16)] / 4, resulting in x = [3 + 5]/4 or x = [3 - 5]/4.
Roots are x = 2 and x = -0.5.
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Example: Finding vertex of y = x^2 - 4x + 3
Use x = -b/(2a) to get x = 2, then y = (2)^2 - 4(2) + 3 = -1, so vertex is (2, -1).
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Example: Solving x^2 - 4 > 0
Roots are x = ±2; test intervals to find x < -2 or x > 2.
Solution is x < -2 or x > 2.
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Example: Sum and product for x^2 - 7x + 12
Sum of roots is 7, product is 12; roots are 3 and 4.
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FOIL method for binomials
A way to multiply two binomials, like (x + a)(x + b) = x^2 + (a+b)x + ab, essential for expanding factored quadratics.
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Quadratic inequality graphing
Graphing y = ax^2 + bx + c and shading regions above or below the x-axis based on the inequality sign.
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Systems with quadratic and linear
Solving a system like y = x^2 + 1 and y = 2x + 1 by substitution, leading to a quadratic equation.
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Discriminant calculation
For ax^2 + bx + c = 0, compute b^2 - 4ac; for example, in x^2 - 4x + 4, it's 16 - 16 = 0.
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Vertex formula
The formula for the vertex x-coordinate is -b/(2a), used to quickly locate the parabola's turning point.
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Quadratic with no real solutions
A quadratic like x^2 + 1 = 0 where the discriminant is negative, meaning the graph doesn't cross the x-axis.
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Quadratic with two real solutions
A quadratic like x^2 - 4 = 0 where the discriminant is positive, resulting in two distinct x-intercepts.
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Quadratic with one real solution
A quadratic like x^2 - 2x + 1 = 0 where the discriminant is zero, touching the x-axis at one point.
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Setting up equations for maximum area
In problems like fencing a rectangle, express area as a quadratic and find the vertex for the maximum.
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Time to reach maximum height
In projectile motion, use t = -b/(2a) from the height equation to find when the object peaks.
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Deriving the quadratic formula
Start with ax^2 + bx + c = 0, divide by a, complete the square, and solve to arrive at x = [-b ± sqrt(b^2 - 4ac)] / (2a).
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Common factoring mistakes
Errors such as incorrect signs in binomials or failing to factor out a greatest common factor first.
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Identifying quadratic from word problem
Recognizing phrases like 'area of a garden' or 'time of flight' that lead to equations like length times width equaling a quadratic.