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Quadratic
Equations

Master solving equations of the form \( ax^2 + bx + c = 0 \) using factoring, the quadratic formula, and completing the square.

SAT Math

What Is a Quadratic Equation?

A quadratic equation is any equation that can be written in the form \( ax^2 + bx + c = 0 \), where \( a \neq 0 \). The highest power of the variable is 2, which gives the equation its characteristic parabolic shape when graphed.

\( a \)
Leading coefficient: controls the width and direction of the parabola
\( b \)
Linear coefficient: shifts the axis of symmetry left or right
\( c \)
Constant term: the y-intercept of the parabola

Graphing: The Parabola

Why Quadratics Matter on the SAT

Quadratic equations appear across multiple question types on the SAT. You will need to solve them to answer questions about:

Algebra
Solving for unknown values, simplifying expressions
Word Problems
Projectile motion, area, revenue optimization
Graphing
Vertex, intercepts, axis of symmetry, transformations

Three Solving Methods

1
Factoring: Rewrite \( ax^2 + bx + c \) as a product of two binomials, then set each factor equal to zero.
2
Quadratic Formula: Use \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find solutions directly from the coefficients.
3
Completing the Square: Rewrite the equation in vertex form \( a(x - h)^2 + k = 0 \) and solve for \( x \).

The Quadratic Formula

This formula gives the exact solutions of any quadratic equation \( ax^2 + bx + c = 0 \). The expression under the square root, \( b^2 - 4ac \), is called the discriminant.

\( \Delta > 0 \)
Two distinct real solutions
\( \Delta = 0 \)
One repeated real solution
\( \Delta < 0 \)
No real solutions (two complex)

The Discriminant: \( b^2 - 4ac \)

Factoring vs. Quadratic Formula

Factoring Quadratic Formula
Speed Fastest (when it works) Moderate
Always works? No: only for factorable equations Yes: works for all quadratics
Best when Coefficients are small integers Coefficients are large or irrational roots
SAT tip Try this first (under 30 seconds) Fall back if factoring fails

Step-by-Step Strategy

1
Set equal to zero: Move all terms to one side so the equation is \( ax^2 + bx + c = 0 \).
2
Identify \( a \), \( b \), \( c \): Read off the coefficients carefully (watch the signs).
3
Try factoring: Look for two numbers that multiply to \( ac \) and add to \( b \).
4
If factoring fails, use the formula: Substitute into \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
5
Check your answers: Substitute each solution back into the original equation to verify.

Example 1: Factoring

1
Find two numbers that multiply to \( 6 \) and add to \( 5 \): those are \( 2 \) and \( 3 \).
2
Factor: \( (x + 2)(x + 3) = 0 \).
3
Set each factor to zero: \( x + 2 = 0 \) or \( x + 3 = 0 \).
4
Solutions: \( x = -2 \) and \( x = -3 \).

Example 2: Quadratic Formula

1
Identify: \( a = 2 \), \( b = -3 \), \( c = -5 \).
2
Discriminant: \( (-3)^2 - 4(2)(-5) = 9 + 40 = \) \( 49 \).
3
Apply the formula: \( x = \frac{3 \pm \sqrt{49}}{4} = \frac{3 \pm 7}{4} \).
4
Solutions: \( x = \frac{10}{4} = \frac{5}{2} \) and \( x = \frac{-4}{4} = -1 \).

Example 3: Completing the Square

1
Move the constant: \( x^2 + 6x = -2 \).
2
Half of \( b \) is \( 3 \). Square it: \( 9 \). Add to both sides: \( x^2 + 6x + 9 = 7 \).
3
Write as a perfect square: \( (x + 3)^2 = 7 \).
4
Take the square root: \( x + 3 = \pm\sqrt{7} \), so \( x = -3 \pm \sqrt{7} \).

Vertex Form and the Vertex

Every quadratic can be rewritten in vertex form: \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.

\( h \)
\( h = \frac{-b}{2a} \): the x-coordinate of the vertex and the axis of symmetry
\( k \)
\( k = f(h) \): the minimum (or maximum) value of the quadratic

Sum and Product of Roots

For \( ax^2 + bx + c = 0 \) with roots \( r_1 \) and \( r_2 \), two powerful identities hold:

Sum of Roots
\( r_1 + r_2 = \frac{-b}{a} \)
Product of Roots
\( r_1 \cdot r_2 = \frac{c}{a} \)

Common Mistakes to Avoid

✕ Sign errors in \( b \)
When \( b \) is negative, \( -b \) becomes positive. Forgetting the double negative in \( \frac{-(-3)}{2a} \) is the most common error.
✕ Forgetting \( \pm \)
The formula gives two solutions. Writing only the \( + \) case means you lose one root and get a wrong answer.
✕ Not setting equal to 0
Factoring \( x^2 + 5x = -6 \) directly is wrong. You must rearrange to \( x^2 + 5x + 6 = 0 \) first.

Method Comparison Summary

Factoring Formula Completing Sq.
Always works No Yes Yes
Speed Fastest Moderate Slowest
Gives vertex No No Yes
SAT use Most questions Backup / hard ones Vertex questions

Question 1 of 5

What are the solutions of \( x^2 - 7x + 12 = 0 \)?

Question 2 of 5

What is the discriminant of \( 3x^2 + 2x + 5 = 0 \), and what does it tell you?

Question 3 of 5

If the sum of the solutions of \( 2x^2 - 10x + k = 0 \) is 5, what is the value of \( k \) if the product of the solutions is 3?

Question 4 of 5

What is the vertex of the parabola \( y = x^2 - 8x + 15 \)?

Question 5 of 5

Solve using the quadratic formula: \( x^2 + 4x + 1 = 0 \)