Find the solutions of $f(x)=x^2+x-12$.

Question

02-11-2026
Mathematics

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Find the solutions of $f(x)=x^2+x-12$.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-02T00:01:44-05:00

Set the function $f(x) = x^2 + x - 12$ to zero to create the equation $x^2 + x - 12 = 0$.
Factor the quadratic expression into $(x + 4)(x - 3) = 0$.
Apply the zero product property to solve $x + 4 = 0$ and $x - 3 = 0$.
The solutions to the equation are $\boxed{x=3, x=-4}$.

Explanation

Analyzing the quadratic function

The given function is $f(x) = x^2 + x - 12$. To find the solutions, we must set the function equal to zero, which gives us the quadratic equation $x^2 + x - 12 = 0$. This is in the form $ax^2 + bx + c = 0$, with coefficients $a=1$, $b=1$, and $c=-12$. We aim to find the values of $x$ that satisfy this equation.

Factoring the quadratic expression

One efficient way to solve this quadratic equation is through factoring. We look for two numbers that multiply to the constant term $c = -12$ and add up to the coefficient of the middle term $b = 1$. Testing factors of $-12$, we find that $4$ and $-3$ satisfy these conditions: $4 \times (-3) = -12$ and $4 + (-3) = 1$. Therefore, the quadratic expression can be factored as $(x + 4)(x - 3) = 0$.

Solving for $x$ using the zero product property

Using the zero product property, if the product of two factors is zero, then at least one of the factors must be zero. This gives us two linear equations: $x + 4 = 0$ and $x - 3 = 0$. Solving these, we get $x = -4$ and $x = 3$.

Verification and final answer

We can verify these solutions using the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Plugging in our values:
\[x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-12)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 48}}{2} = \frac{-1 \pm \sqrt{49}}{2} = \frac{-1 \pm 7}{2}.\]
This yields $x = \frac{-1 + 7}{2} = 3$ and $x = \frac{-1 - 7}{2} = -4$. Both methods confirm that the solutions to $f(x) = 0$ are $x = -4$ and $x = 3$. The final solutions are $\boxed{x=3, x=-4}$.

Examples

Quadratic functions are fundamental for modeling physical phenomena, such as the path of a projectile. If a ball is thrown into the air, its height over time might be represented by a quadratic equation. Finding where the function equals zero helps determine the specific times when the ball reaches the ground. This knowledge is essential in engineering, physics, and any field involving motion, optimization, or balanced structures where trajectories need to be calculated.