Factor: $x^{2} - 11x + 28$

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31-10-2026
Mathematics

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Factor: $x^{2} - 11x + 28$

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-10-31T21:13:33-04:00

Identify coefficients of the quadratic expression $x^2 - 11x + 28$, where $b = -11$ and $c = 28$.
Determine two negative factors of $28$ that sum to $-11$, which are $-4$ and $-7$.
Write the expression in factored form as $(x - 4)(x - 7)$.
The final result is $\boxed{(x - 4)(x - 7)}$.

Explanation

Problem Analysis

We are given the quadratic expression $x^2 - 11x + 28$. This is a quadratic trinomial of the form $ax^2 + bx + c$, where the leading coefficient $a = 1$, the coefficient $b = -11$, and the constant term $c = 28$. Our goal is to factor this expression into the product of two linear binomials of the form $(x + p)(x + q)$, where $p$ and $q$ are constants. Expanding this product, we get $x^2 + (p + q)x + pq$. Thus, we need to find two numbers $p$ and $q$ such that their sum $p + q = -11$ and their product $pq = 28$.

Finding the Factors

To find the values of $p$ and $q$, we look for factors of the constant term $c = 28$. Since the product $pq = 28$ is positive and the sum $p + q = -11$ is negative, both $p$ and $q$ must be negative numbers. Let us list the pairs of negative factors of $28$:

$-1$ and $-28$: sum is $-1 + (-28) = -29$
$-2$ and $-14$: sum is $-2 + (-14) = -16$
$-4$ and $-7$: sum is $-4 + (-7) = -11$

The pair that satisfies the condition that their sum is $-11$ is $-4$ and $-7$.

Final Factoring and Verification

Now that we have identified the constants $p = -4$ and $q = -7$, we can substitute them into the factored form $(x + p)(x + q)$. This gives us the expression $(x - 4)(x - 7)$. We can verify this by expanding the product: $(x - 4)(x - 7) = x \cdot x + x \cdot (-7) + (-4) \cdot x + (-4) \cdot (-7) = x^2 - 7x - 4x + 28 = x^2 - 11x + 28$. Since this matches our original expression, the factoring is correct. The final factored form is $\boxed{(x - 4)(x - 7)}$.

Examples

Factoring quadratic expressions is a fundamental skill in algebra used to solve problems across many disciplines. For example, if you are designing the layout of a rectangular garden and you know its area is represented by the quadratic equation $x^2 - 11x + 28 = 0$, factoring helps you find the dimensions of the garden. By setting the factors $(x - 4) = 0$ and $(x - 7) = 0$, you can quickly determine that the side lengths must be $4$ units and $7$ units. This simplifies complex problems involving areas, projectile motion, and optimization into manageable linear equations.