How do you solve a quadratic equation using the quadratic formula?

Conditions

• The equation must be in standard quadratic form: $ax^2 + bx + c = 0$.
• The leading coefficient $a$ must not be zero ($a \neq 0$).

Reasoning, step by step

1. Rewrite the equation so all terms are on one side and zero is on the other.
2. Identify the coefficients $a$, $b$, and $c$ from the standard form.
3. Substitute $a$, $b$, and $c$ into the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
4. Calculate the value of the discriminant $b^2 - 4ac$ under the square root.
5. Evaluate the principal square root of the discriminant.
6. Split the expression into two separate cases using the $+$ and $-$ signs.
7. Perform the arithmetic for each case to get two fractions.
8. Reduce the fractions to their simplest form to find the final solutions.

Example

For the equation $2x^2 + 3x - 2 = 0$, the coefficients are $a=2$, $b=3$, and $c=-2$. Substituting these into the formula gives $x = \frac{-(3) \pm \sqrt{(3)^2 - 4(2)(-2)}}{2(2)}$. Simplifying the discriminant yields $\sqrt{9 + 16} = \sqrt{25} = 5$, so the expression becomes $x = \frac{-3 \pm 5}{4}$. Splitting this gives $x = \frac{2}{4} = \frac{1}{2}$ and $x = \frac{-8}{4} = -2$.

Common misconceptions

• Misidentifying the sign of the constant term $c$; for example, treating $c$ as positive 2 when the equation has '-2', which incorrectly changes the discriminant calculation.
• Forgetting to split the $\pm$ symbol into two separate calculations, leading to only one solution instead of two.
• Assuming the quadratic formula can be applied directly without first rearranging the equation into standard form.

Watch the explanation

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