LEMMA

Polynomials & quadratics

Quadratic formula

ALEKS placement

Read the signed coefficients, then use one formula to find the quadratic's solutions.

What this covers

x=2a−b±√b2−4ac​​
  • Write it as ax² + bx + c = 0 and read off a, b, c with their signs.
  • Write the whole formula out before substituting, then replace a, b, c one at a time, each in parentheses.
  • When b is negative, −b turns positive and b² is positive too: b = −4 gives −b = 4 and b² = 16.
  • Compute the discriminant b² - 4ac first. Positive means two real roots, zero means one, negative means none real.
  • Split the ± into two fractions, one with plus and one with minus. Simplify the radical, then reduce the whole fraction.
  • Check each distinct solution in the original equation. State when there are no real solutions.

The formula works for every quadratic with a nonzero squared-term coefficient. It is useful when factoring is difficult. Completing the square can find the same solutions and explains where this formula comes from.

Worked example

Worked example

Solve 2x² − 4x − 3 = 0

  1. a=2b=−4c=−3​Read the coefficients from an equation with zero on the right. Keep each sign attached to its number.
  2. D=(−4)2−4(2)(−3)The discriminant is the expression under the square root. Substitute each signed coefficient in parentheses.
  3. D=16+24=40Squaring negative four gives a positive value. Subtracting the negative product adds twenty-four. A positive discriminant means two real solutions.
  4. x=44±√40​The negative of the linear coefficient is positive four. Twice the squared-term coefficient is the denominator for the whole numerator.
  5. x=44±2√10​Forty contains a square factor of four. Its square root contributes a factor of two outside the radical.
  6. x=22±√10​Every part of the fraction shares a factor of 2.
  7. x=22+√10​orx=22−√10​​The plus-or-minus sign represents two possibilities. Keep both solutions in exact form.
Another worked example

Solve 4x² − 9x + 2 = 0 exactly.

  1. D=(−9)2−4(4)(2)=49Here a = 4, b = −9, c = 2. The discriminant 81 − 32 = 49 is a perfect square, so the roots are rational.
  2. x=89±7​Put −b = 9, √49 = 7, and 2a = 8 into the formula. The 8 divides the whole numerator.
  3. x=2orx=41​(9 + 7)/8 = 2 and (9 − 7)/8 = 2/8 = 1/4.

A common mistake

A mistake Lemma catches

From D=(−4)2−4(2)(−3)

D=−16+24D=16+24

“(−4)² squares the whole −4: (−4)(−4) = 16. Writing −16 squares only the 4 and keeps the minus outside: −4² = −(4²) = −16. Keep the brackets whenever b is negative.”

Try one

Sample problem

Solve using the quadratic formula.

x2+7x+6=0

Practice quadratic formula free

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