Solve a quadratic equation using the quadratic formula

Key Notes :

Understanding the Quadratic Formula

  • The quadratic formula is a general formula used to solve any quadratic equation of the form ax² + bx + c = 0.
  • It is derived from completing the square and provides a direct method to find the solutions.

The quadratic formula is

x = – b ± √b2 -4ac / 2a

where a, b, and c are the coefficients of the quadratic equation.

Steps to Solve Using the Quadratic Formula

  1. Identify the Coefficients: Determine the values of a, b, and c from the given quadratic equation.
  2. Substitute into the Formula: Substitute the values of a, b, and c into the quadratic formula.
  3. Simplify and Solve: Evaluate the expression under the square root (discriminant), take the square root, and simplify the resulting expression.
  4. Find the Solutions: Solve for x by considering both the positive and negative square root values.

Example

Solve: 2x² – 5x + 3 = 0

  • Step 1: a = 2, b = -5, c = 3
  • Step 2: Substitute into the formula: x = (-(-5) ± √((-5)² – 4(2)(3))) / (2(2))
  • Step 3: Simplify: x = (5 ± √1) / 4
  • Step 4: Solve: x = (5 + 1) / 4 = 3/2 or x = (5 – 1) / 4 = 1

Key Points

  • The quadratic formula is a versatile tool for solving quadratic equations.
  • It works for any quadratic equation, regardless of whether it can be factored or not.
  • The discriminant (b² – 4ac) determines the nature of the solutions:
    • Positive: Two distinct real solutions
    • Zero: One repeated real solution
    • Negative: Two complex conjugate solutions
  • Practice using the quadratic formula to improve your skills and understanding.

Learn with an example

Solve using the quadratic formula.

3j2 + 5j + 1 = 0

Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.

j = __________ or j = _____________

Use the quadratic formula to solve 3j2 + 5j + 1 = 0.

j = -b± √b2 − 4ac / 2a

j = -5 ± √52 -4(3)(1) / 2(3) Plug in a = 3, b = 5, and c = 1

j = -5 ± √25-12 / 6 Multiply

j = -5 ± √13 / 6 Subtract

j = -5 +√13 / 6 or   j = -5-√13 / 6 Split ± into + or –

j  ≈ -0.23  or  j ≈ -1.43 Simplify and round to the nearest hundredth

Solve using the quadratic formula.

5u2 − 5u − 4 = 0

Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.

u = _____________ or u = ______________

Use the quadratic formula to solve 5u2 − 5u − 4 = 0.

u = -b± √b2 − 4ac / 2a

j = -(-5) ± √(-5)2 -4(5)(-4) / 2(5) Plug in a = 5, b = -5, and c = -4

u = 5 ± √25+80 / 10 Multiply

j = 5 ± √105 / 10 Add

j = 5 +√105 / 10 or   j = 5-√105 / 10 Split ± into + or –

u  ≈  1.52  or  u ≈ -0.52 Simplify and round to the nearest hundredth

Solve using the quadratic formula.

j2 + 6j + 9 = 0

Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.

j = __________ or j = _____________

Use the quadratic formula to solve j2 + 6j + 9 = 0.

j = -b± √b2 − 4ac / 2a

j = -6 ± √62 -4(1)(9) / 2(1) Plug in a = 1, b = 6, and c = 9

j = -6 ± √36-36 / 2 Multiply

j = -6 ± √0 / 2 Subtract

j = -6 +√0 / 2 or   j = -6-√0 / 2 Split ± into + or –

j  = -3  or  j = -3 Simplify

The two solutions are the same, so they should be written as a single solution : j = -3.

let’s practice!