Quadratic formula

Japanese school year: Junior high 3

What you learn

Learn how to solve quadratic equations using the quadratic formula derived from completing the square. This formula enables students to reliably find solutions even when an equation cannot be easily factored. Prior understanding of square root properties and basic algebraic operations ensures smooth learning.

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Key points

This formula gives the solutions $x$ for any quadratic equation, even if it cannot be factored. Simply substitute the coefficients $a$ , $b$ , and $c$ (with $a \ne 0$ ) into the formula.

ax^2 + bx + c = 0 \implies x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Use this simpler formula when the coefficient of $x$ is an even number $2b'$ . Using half of that coefficient as $b'$ avoids mistakes and saves simplifying steps.

ax^2 + 2b'x + c = 0 \implies x = \frac{-b' \pm \sqrt{b'^2 - ac}}{a}

This is an important check when using the quadratic formula. If the value inside the square root ( $b^2 - 4ac$ ) is negative, there are no real solutions.

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Practice 1Coming soon