Factor and solve: $6x^2 + 11x - 10 = 0$

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braemarie braemarie

02-11-2026
Mathematics

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Factor and solve: $6x^2 + 11x - 10 = 0$

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-02T20:57:29-05:00

Rewrite the middle term $11x$ as $15x - 4x$ because $15 \times (-4) = -60$ and $15 + (-4) = 11$.
Factor by grouping to obtain $(3x - 2)(2x + 5) = 0$.
Set each factor equal to zero to find the roots.
The solutions are $\boxed{x = \frac{2}{3}, -\frac{5}{2}}$.

Explanation

Analyzing the Quadratic Equation

The given equation is $6x^2 + 11x - 10 = 0$. This is a quadratic equation of the form $ax^2 + bx + c = 0$, where $a = 6$, $b = 11$, and $c = -10$. Our goal is to factor the quadratic expression and solve for $x$. To factor this expression, we look for two numbers that multiply to $a \times c = 6 \times (-10) = -60$ and add up to $b = 11$.

Finding Factors and Rewriting the Expression

We need to find two numbers $m$ and $n$ such that $m \times n = -60$ and $m + n = 11$. Testing factors of $-60$, we observe that $15$ and $-4$ satisfy both conditions:
$15 \times (-4) = -60$ and $15 + (-4) = 11$.
With these values, we can rewrite the middle term $11x$ as $15x - 4x$:
$6x^2 + 15x - 4x - 10 = 0$.

Factoring by Grouping

Now, we perform factoring by grouping. We group the terms in pairs:
$(6x^2 + 15x) + (-4x - 10) = 0$.
Next, we factor out the greatest common factor from each pair:
$3x(2x + 5) - 2(2x + 5) = 0$.
Since both terms share the common factor $(2x + 5)$, we can factor it out:
$(3x - 2)(2x + 5) = 0$.

Solving for x

Using the zero-product property, we set each factor to zero to solve for $x$:
1) $3x - 2 = 0 \implies 3x = 2 \implies x = \frac{2}{3}$
2) $2x + 5 = 0 \implies 2x = -5 \implies x = -\frac{5}{2}$
The solutions for the equation are $x = \frac{2}{3}$ and $x = -\frac{5}{2}$. Thus, the factored form is $(3x-2)(2x+5) = 0$ and the solutions are $\boxed{x = \frac{2}{3}, -\frac{5}{2}}$.

Examples

Quadratic equations like this are frequently used in physics and engineering to model projectile motion. For example, if you were launching an object from a certain height, the path of the object can often be described by a quadratic equation where the variable represents time and the result represents the height of the object. Solving for the roots tells you exactly when the object reaches certain heights, such as when it hits the ground (height = 0). Understanding how to factor these equations is a fundamental skill for solving many real-world kinematic problems.