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Algebra 2 practice

How to Use the Quadratic Formula

To use the quadratic formula, get the equation into ax2+bx+c=0ax^2 + bx + c = 0 form, identify aa, bb, cc with their signs, and substitute into x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. Below: a worked example step by step, the sign traps that cause most wrong answers, and practice with every step explained.

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What you'll practice

The quadratic formula solves every equation of the form ax2+bx+c=0ax^2 + bx + c = 0, including the ones factoring can't touch. These questions cover identifying aa, bb, cc (signs included), simplifying the discriminant b2−4acb^2 - 4ac, and using the discriminant to predict how many real solutions exist before solving.

Worked example

Solve x2−4x−5=0x^2 - 4x - 5 = 0.

  1. Identify a=1a = 1, b=−4b = -4, c=−5c = -5.
  2. Discriminant: b2−4ac=(−4)2−4(1)(−5)=16+20=36b^2 - 4ac = (-4)^2 - 4(1)(-5) = 16 + 20 = 36.
  3. x=−(−4)±362(1)=4±62x = \frac{-(-4) \pm \sqrt{36}}{2(1)} = \frac{4 \pm 6}{2}.
  4. x=102=5x = \frac{10}{2} = 5 or x=−22=−1x = \frac{-2}{2} = -1. Check: 25−20−5=025 - 20 - 5 = 0. ✓

Common mistakes

  • Dropping the sign of bb: for b=−4b = -4, the formula's −b-b is +4+4. Write the substitution in parentheses before simplifying.
  • Discriminant sign error: −4ac-4ac with a negative cc becomes positive — 16+2016 + 20, not 16−2016 - 20.
  • Dividing only one term by 2a2a: BOTH the −b-b and the ± \pm\sqrt{\ } part sit over 2a2a.

Try a few

1. Solve x2+2x−8=0x^2 + 2x - 8 = 0.

  • x=2x = 2 or x=−4x = -4
  • x=−2x = -2 or x=4x = 4
  • x=1x = 1 or x=−8x = -8
  • x=2x = 2 or x=4x = 4

2. Solve 2x2−3x−2=02x^2 - 3x - 2 = 0.

  • x=2x = 2 or x=−12x = -\frac{1}{2}
  • x=−2x = -2 or x=12x = \frac{1}{2}
  • x=1x = 1 or x=−2x = -2
  • x=2x = 2 or x=12x = \frac{1}{2}

3. What does the discriminant of x2−6x+9=0x^2 - 6x + 9 = 0 tell you?

  • b2−4ac=0b^2-4ac = 0: exactly one repeated real solution
  • b2−4ac>0b^2-4ac > 0: two distinct real solutions
  • b2−4ac<0b^2-4ac < 0: no real solutions
  • The discriminant can't be found without solving

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FAQ

How do you use the quadratic formula step by step?
Write the equation as ax² + bx + c = 0, identify a, b, and c with their signs, substitute into x = (−b ± √(b² − 4ac)) / 2a, simplify under the square root first, then split the ± into your two solutions.
When should I use the quadratic formula instead of factoring?
Whenever factoring isn't quick — the formula works on every quadratic. If the discriminant isn't a perfect square, the equation didn't factor nicely anyway.
What does the discriminant tell you?
b² − 4ac predicts the solutions before you solve: positive means two real solutions, zero means one repeated solution, negative means no real solutions.
Is Cramzoo free?
Yes — this skill is one of the free ones. Every unit opens with free skills, no account needed. Premium ($6/month or $30/year) unlocks every skill plus the parent progress dashboard.
How is this different from IXL?
No punishing score. A wrong answer gives you a step-by-step explanation, not a number that drops — and every question is double-checked for accuracy before it reaches you.

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