Free video lesson

How to Find Both Solutions of a Quadratic on the SAT

When a quadratic factors, you don't need the formula — the roots fall right out. Here's how to factor fast and read the roots off the page, plus the Desmos shortcut for roots and the vertex.

Math · Nonlinear equations · one variable2:10Published July 5, 2026

On YouTube: SAT Quadratics: Check for Two Solutions

What this lesson covers

  • Factor x² + bx + c: two numbers that multiply to c and add to b
  • Roots are where y = 0 — set each factor to zero
  • Desmos: graph it, click the x-intercepts (roots) and the bottom (vertex)
  • Practice: x² − x − 12
  • The three sign traps that cost points

Questions worked in the video

  1. 0:29y = x² − 3x − 10 — factor it and find the roots.
  2. 1:09y = x² − x − 12 — factor it and find both roots.

Worked examples

The questions the video works, written out: the setup, each step, the answer and the trap.

0:29Example 1

The graph of y=x2−3x−10y = x^2 - 3x - 10 is a parabola. Where does it cross the x-axis? In other words, factor x2−3x−10x^2 - 3x - 10 and find the roots.

  1. The x-axis is where y = 0, so solve x2−3x−10=0x^2 - 3x - 10 = 0. The leading coefficient is 1, so look for two numbers that multiply to the last term, -10, and add to the middle coefficient, -3.
  2. Pairs that multiply to -10: 1 and -10, -1 and 10, 2 and -5, -2 and 5. Only 2 and -5 add to -3.
  3. Write the factors: x2−3x−10=(x+2)(x−5)x^2 - 3x - 10 = (x + 2)(x - 5).
  4. Set each factor to zero: x + 2 = 0 gives x = -2, and x - 5 = 0 gives x = 5. The parabola crosses at (-2, 0) and (5, 0).
  5. Check: (−2)2−3(−2)−10=4+6−10=0(-2)^2 - 3(-2) - 10 = 4 + 6 - 10 = 0 and 52−3(5)−10=25−15−10=05^2 - 3(5) - 10 = 25 - 15 - 10 = 0.

Answer: x = -2 and x = 5

A product is zero only when one of its factors is zero, so once the quadratic is written as two factors the roots fall right out with no formula. The trap is flipping a sign at the end: (x - 5) has root +5, not -5, and (x + 2) has root -2. A student who answers 5 and 2, or -5 and 2, has read the factor instead of solving it.

0:49Example 2

For the same parabola y=x2−3x−10y = x^2 - 3x - 10, find the vertex, the lowest point of the graph.

  1. A parabola is symmetric, so the vertex sits exactly halfway between the two roots: x=−2+52=32=1.5\displaystyle x = \frac{-2 + 5}{2} = \frac{3}{2} = 1.5. The formula x=−b2a=32\displaystyle x = -\frac{b}{2a} = \frac{3}{2} gives the same thing.
  2. Plug x = 1.5 back in for the y-coordinate: y=(1.5)2−3(1.5)−10=2.25−4.5−10=−12.25y = (1.5)^2 - 3(1.5) - 10 = 2.25 - 4.5 - 10 = -12.25.
  3. The vertex is (1.5, -12.25). Since the x2x^2 term is positive the parabola opens upward, so this is its minimum.
  4. Check in Desmos: type y=x2−3x−10y = x^2 - 3x - 10, the parabola appears, and clicking the bottom point reads (1.5, -12.25); clicking the two crossings reads (-2, 0) and (5, 0).

Answer: vertex (1.5, -12.25)

The roots do double duty: they are the x-intercepts, and their midpoint is the axis of symmetry, which is where the vertex lives. The trap is reporting the y-intercept (0, -10) as the vertex because it is the easy point to compute; the y-intercept is where x = 0, not the bottom of the curve.

1:09Example 3

Factor y=x2−x−12y = x^2 - x - 12 and find both roots.

  1. Set y = 0: x2−x−12=0x^2 - x - 12 = 0. The middle coefficient is -1, so find two numbers that multiply to -12 and add to -1.
  2. Pairs for -12: 1 and -12, 2 and -6, 3 and -4, plus their sign flips. 3 and -4 multiply to -12 and add to -1.
  3. So x2−x−12=(x+3)(x−4)x^2 - x - 12 = (x + 3)(x - 4). Set each factor to zero: x = -3 and x = 4.
  4. Check: (−3)2−(−3)−12=9+3−12=0(-3)^2 - (-3) - 12 = 9 + 3 - 12 = 0 and 42−4−12=16−4−12=04^2 - 4 - 12 = 16 - 4 - 12 = 0.

Answer: x = -3 and x = 4

The pair must multiply to the constant and add to the middle number; because the constant is negative the two numbers have opposite signs, and the larger one carries the sign of the middle term, here the -4. The trap is choosing -3 and 4, which multiply to -12 but add to +1, giving (x - 3)(x + 4) and the wrong roots 3 and -4.

Chapters

Lesson transcript

The narration of the video, word for word, under its chapter headings.

0:00Skip the formula

Welcome to SAT Climb. Some quadratics you factor in seconds — no formula needed. Today: spot the ones that factor, and read their roots right off the page.

0:16The setup: a parabola

Here's a parabola: y = x² − 3x − 10. The question the SAT loves — where does it cross the x-axis? Those crossings are the roots.

0:29Factor to find roots

Factor it. Find two numbers that multiply to −10 and add to −3: −5 and +2. So it's (x − 5)(x + 2). Set each to zero — the roots are 5 and −2.

0:49Roots & vertex in Desmos

On the Digital SAT, let Desmos do it. Type the equation and the parabola appears. Click each x-intercept for the exact roots, −2 and 5. Click the bottom for the vertex.

1:09Your turn

Your turn. y = x² − x − 12. Factor it, and find both roots. Type it in Desmos if you want to check.

1:35Three traps

Three traps. Signs backwards — it multiplies to the last term, adds to the middle. Forgetting to set each factor to zero. And flipping a root's sign: (x − 5) gives a root of +5.

1:56Recap

Quadratics: solved. When it factors, the roots are right there. Start free at satclimb.com.

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