Split middle term and factorize X square + 32 x minus 420 is equal 0

Answer: Factored form: $(x - 10)(x + 42) = 0$ ; Roots: $x = 10, -42$

Step-by-step solution

Step 1: Identify coefficients and target product

For the quadratic equation in standard form $a x^2 + b x + c = 0$ , we find two integers whose sum equals the middle term coefficient $b = 32$ and whose product equals $a \cdot c = 1 \cdot (-420) = -420$ .

Step 2: Find the factor pair

We test factor pairs of $420$ with opposite signs having a difference of $32$ . The factors $42$ and $-10$ satisfy both conditions: their product is $-420$ and their sum is $42 - 10 = 32$ .

Step 3: Split the middle term and group terms

We rewrite the middle term $32x$ as $42x - 10x$ , splitting the quadratic equation into four terms for grouping.

Step 4: Factor by grouping

Factoring $x$ from the first two terms gives $x(x + 42)$ , and factoring $-10$ from the last two terms gives $-10(x + 42)$ . Taking the common binomial factor $(x + 42)$ outside gives $(x - 10)(x + 42) = 0$ .

Step 5: Solve for the roots

Applying the zero product rule, either $x - 10 = 0$ or $x + 42 = 0$ . This yields the two solutions $x = 10$ and $x = -42$ .

Solve your own maths question free →