Quadratic Equation Calculator

Published on: December 8, 2024
Final Answer: Free • Full Steps: Plus

This Quadratic Equation Calculator solves an equation with an x-squared term in it, and shows the working line by line. The equation is first moved into standard form so the coefficients a, b and c can be read straight off it. The discriminant is worked out before the formula is used, because its sign is what decides whether there are two solutions, one, or none that are real. Following the same pattern every time makes each coefficient easy to trace.

Step-by-step method

  1. Set up the problem.
  2. Rewrite the equation in standard form.
  3. Identify the coefficients a, b and c.
  4. Write the formula for the discriminant.
  5. Substitute the coefficients into the discriminant.
  6. Work out the discriminant.
  7. Write the quadratic formula.
  8. Substitute the coefficients into the quadratic formula.
  9. Simplify to get the solutions.

Formulas:

\(D = b^{2} - 4ac\)
\(x = \frac{-b \pm \sqrt{D}}{2a}\)
See Example 1 Hide Example 1

Problem

\(2x^{2}+3x=5\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Rewrite the equation in standard form.
  3. Identify the coefficients a, b and c.
  4. Write the formula for the discriminant.
  5. Substitute the coefficients into the discriminant.
  6. Work out the discriminant.
  7. Write the quadratic formula.
  8. Substitute the coefficients into the quadratic formula.
  9. Simplify to get the solutions.

Step 1

Step 1 - Set up the problem.

In this problem: The equation to solve is \(2x^{2}+3x=5\), and the unknown is \(x\).

\(2x^{2}+3x=5\)

Step 2

Step 2 - Rewrite the equation in standard form.

In this problem: Moving every term to the left of the equals sign turns \(2x^{2}+3x=5\) into \(2 x^{2} + 3 x - 5 = 0\).

\(2 x^{2} + 3 x - 5 = 0\)

Step 3

Step 3 - Identify the coefficients a, b and c.

In this problem: Reading the coefficients off \(ax^{2} + bx + c = 0\) gives \(a = 2\), \(b = 3\) and \(c = -5\).

\(a = 2,\; b = 3,\; c = -5\)

Step 4

Step 4 - Write the formula for the discriminant.

In this problem: The discriminant is the part under the square root, and its sign is what decides how many real solutions the equation has.

\(D = b^{2} - 4ac\)

Step 5

Step 5 - Substitute the coefficients into the discriminant.

In this problem: Replacing \(a\), \(b\) and \(c\) with \(2\), \(3\) and \(-5\) gives \(D = 3^{2} - 4 \cdot 2 \cdot \left(-5\right)\).

\(D = 3^{2} - 4 \cdot 2 \cdot \left(-5\right)\)

Step 6

Step 6 - Work out the discriminant.

In this problem: Squaring \(3\) and taking away \(4 \cdot 2 \cdot \left(-5\right)\) leaves \(49\).

\(D = 49\)

Step 7

Step 7 - Write the quadratic formula.

In this problem: The quadratic formula undoes the squaring, and the \(\pm\) is what produces the two solutions.

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Step 8

Step 8 - Substitute the coefficients into the quadratic formula.

In this problem: Replacing \(b\) with \(3\), \(a\) with \(2\) and \(D\) with \(49\) gives \(x = \frac{-3 \pm \sqrt{49}}{4}\).

\(x = \frac{-3 \pm \sqrt{49}}{4}\)

Step 9

Step 9 - Simplify to get the solutions.

In this problem: Taking the \(+\) branch gives \(1\) and the \(-\) branch gives \(- \frac{5}{2}\).

\(x = 1,\; x = - \frac{5}{2}\)

Final Answer

\(x = 1,\; x = - \frac{5}{2}\)
See Example 2 Hide Example 2

Problem

\(9x^{2}-6x+1=0\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Rewrite the equation in standard form.
  3. Identify the coefficients a, b and c.
  4. Write the formula for the discriminant.
  5. Substitute the coefficients into the discriminant.
  6. Work out the discriminant.
  7. Write the quadratic formula.
  8. Substitute the coefficients into the quadratic formula.
  9. Simplify to get the solutions.

Step 1

Step 1 - Set up the problem.

In this problem: The equation to solve is \(9x^{2}-6x+1=0\), and the unknown is \(x\).

\(9x^{2}-6x+1=0\)

Step 2

Step 2 - Rewrite the equation in standard form.

In this problem: Everything is already on the left of the equals sign with \(0\) on the right, so the equation is unchanged.

\(9 x^{2} - 6 x + 1 = 0\)

Step 3

Step 3 - Identify the coefficients a, b and c.

In this problem: Reading the coefficients off \(ax^{2} + bx + c = 0\) gives \(a = 9\), \(b = -6\) and \(c = 1\).

\(a = 9,\; b = -6,\; c = 1\)

Step 4

Step 4 - Write the formula for the discriminant.

In this problem: The discriminant is the part under the square root, and its sign is what decides how many real solutions the equation has.

\(D = b^{2} - 4ac\)

Step 5

Step 5 - Substitute the coefficients into the discriminant.

In this problem: Replacing \(a\), \(b\) and \(c\) with \(9\), \(-6\) and \(1\) gives \(D = \left(-6\right)^{2} - 4 \cdot 9 \cdot 1\).

\(D = \left(-6\right)^{2} - 4 \cdot 9 \cdot 1\)

Step 6

Step 6 - Work out the discriminant.

In this problem: Squaring \(-6\) and taking away \(4 \cdot 9 \cdot 1\) leaves \(0\).

\(D = 0\)

Step 7

Step 7 - Write the quadratic formula.

In this problem: The quadratic formula undoes the squaring, and the \(\pm\) is what produces the two solutions.

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Step 8

Step 8 - Substitute the coefficients into the quadratic formula.

In this problem: Replacing \(b\) with \(-6\), \(a\) with \(9\) and \(D\) with \(0\) gives \(x = \frac{6 \pm \sqrt{0}}{18}\).

\(x = \frac{6 \pm \sqrt{0}}{18}\)

Step 9

Step 9 - Simplify to get the solutions.

In this problem: The two branches of the \(\pm\) give the same value, so there is one repeated solution, \(\frac{1}{3}\).

\(x = \frac{1}{3}\)

Final Answer

\(x = \frac{1}{3}\)

Example Problems

Click one to load it into the box and solve it.