This unit is ≈18% of the SAT, across 6 lessons. Full syllabus
Lesson 2 of 6 · Advanced math
Quadratic equations
6 min read · about 1 h 40 min with practice3 quick checks≈4% of the testCore: Core: tested on most papers
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Quadratic equations appear on every SAT: as quick solves early in module 1 and as parameter puzzles (discriminant, sum of roots, “exactly one solution”) late in the harder module 2. Four solving methods, the discriminant and the sum-and-product shortcuts cover nearly all of them.
By the end you’ll be able to
Solve quadratics by factoring, taking square roots, completing the square and the quadratic formula
Use the discriminant to determine the number of real solutions and to find parameter values
Use the sum of roots and product of roots relationships
Create and solve quadratic equations from contexts
What the exam asks
Typical stem
What you actually do
What is the positive (or negative) solution to the given equation?
Solve, then pick the root that meets the condition
Which of the following gives the solutions to the given equation?
Quadratic formula or completing the square; choices contain radicals
How many distinct real solutions does the given equation have?
Compute the discriminant b2−4ac
What is the sum (or product) of the solutions?
Use − or . Don’t solve at all
vii.Check your understanding
3 questions on quadratic equations. Every option is explained once you answer.
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First move every term to one side and divide out any common factor (such as −16 in a projectile equation).
Factoring and the zero product property
If AB=0, then A=0 or B=0. For x2+bx+c, find two numbers that multiply to c and add to b. For ax2+bx+c, use the ac method. With 2x2−13x+15, ac=30, and −3 and −10 multiply to 30 and add to −13:
2x2−3x−10x+15=x(2x−3)−5(2x−3)=(2x−3)(x−5)
Watch for special patterns: x2−7x=x(x−7) (common factor) and x2−49=(x−7)(x+7) (difference of squares).
Completing the square
Get the x-terms alone: x2+bx=−c (divide by a first if a=1).
Add (2b)2 to both sides.
Write the left side as (x+2b)2 and take square roots.
SAT items that say “in the form (x−p)2=q” test exactly this. For x2+bx+c=0 the result is always q=(2b)2−c.
The quadratic formula and the discriminant
x=2a−b±b2−4ac
The discriminant D=b2−4ac tells you how many real solutions there are before you solve anything:
Discriminant
Real solutions
Graph of y=ax2+bx+c
D>0
Two distinct
Crosses the x-axis twice
D=0
Exactly one (a double root at x=−2ab)
Touches the -axis at its vertex
D<0
None
Never meets the x-axis
When the leading coefficient is a parameter, check it separately. If k=0 in kx2−6x+3=0, the equation becomes linear and has exactly one solution. So “two distinct solutions” needs D>0andk=0, while “exactly one solution” can come from D=0 or from k=0.
Sum and product of the solutions
For ax2+bx+c=0 with solutions r and s:
r+s=−abrs=ac
These relationships answer “find the other solution”, “find r1+s1=rsr+s” and “find r2+s2=(r+s)2” in one or two lines. The distance between the two solutions is ∣a∣D.
Building an equation from a context
Name the unknown, write each quantity in terms of it, solve, and reject impossible roots (negative lengths or times). Then reread what the question asks for: the width or the length, the time or the height?
Worked examples
Exam technique
Classify in five seconds. Solve, count, sum/product, or parameter? Counting and sum/product items never need the roots.
Let Desmos solve. Typing x^2-5x-14=0 into Desmos draws vertical lines at the solutions, so checking a hand solution takes about 10 seconds.
Use sliders for parameters. Graph y=kx2−6x+3 with a slider and watch the parabola cross the axis, touch it or lift off it.
Back-solve context MCQs by plugging the choices into the story.
Mind the SPR box. Include the minus sign, and enter fractions exactly (13/2) or as decimals (6.5).
Common mistakes
Quick recap
Set the equation equal to 0 first. Then use square roots for (square) = number; factoring for friendly roots; completing the square or the formula otherwise; Desmos to check.
Discriminant: positive means two solutions, zero means one, negative means none. A parameter on x2 also needs a separate check at 0.
Sum of solutions −ab, product ac: use them for “other root”, reciprocal-sum and r2+s2 questions.
In contexts, reject impossible roots and answer the exact quantity asked.
Hard items add a second condition (positive k, a negative solution) to eliminate one of two answers.