Solve the Quadratic Equation: x²-7x+12=0 Step by Step

Q: How do I know which two numbers to pick for factoring?

Look for two numbers that multiply to give the constant term (12) and add to give the coefficient of x (-7). List factor pairs of 12: (1,12), (2,6), (3,4), then check which pair adds to -7.

Q: Why do I get two different answers?

Quadratic equations usually have two solutions because a parabola crosses the x-axis at two points. Both x = 3 and x = 4 make the original equation equal zero.

Q: How can I check my factoring is correct?

Expand your factored form using FOIL: (x-3)(x-4) = $ x^2 - 4x - 3x + 12 = x^2 - 7x + 12 $. If it matches the original, you're right!

Q: What does it mean when (x-3)(x-4) = 0?

This uses the Zero Product Property: if two things multiply to zero, at least one must be zero. So either x-3=0 (giving x=3) or x-4=0 (giving x=4).

Quadratic Factoring with Integer Coefficients

$x^2-7x+12=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 We'll break it down using trinomial, we'll look at the coefficients

00:08 We want to find 2 numbers whose sum equals B (-7)

00:15 and their product equals C (12)

00:23 These are the matching numbers, we'll substitute in parentheses

00:33 We'll find what zeros each factor

00:43 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$x^2-7x+12=0$

Step-by-step solution

Let's observe that the given equation:

$x^2-7x+12=0$ is a quadratic equation that can be solved using quick factoring:

$x^2-7x+12=0 \longleftrightarrow\begin{cases}\underline{?}\cdot\underline{?}=12\\ \underline{?}+\underline{?}=-7\end{cases}\\ \downarrow\\ (x-3)(x-4)=0$ and therefore we get two simpler equations from which we can extract the solution:

$(x-3)(x-4)=0 \\ \downarrow\\ x-3=0\rightarrow\boxed{x=3}\\ x-4=0\rightarrow\boxed{x=4}\\ \boxed{x=3,4}$ Therefore, the correct answer is answer A.

Final Answer

$x=3,x=4$

Key Points to Remember

Essential concepts to master this topic

Rule: Find two numbers that multiply to c and add to b
Technique: For $x^2-7x+12$ , find factors of 12 that add to -7: -3 and -4
Check: Verify (x-3)(x-4) = $x^2-7x+12$ by expanding ✓

Common Mistakes

Avoid these frequent errors

Getting the signs wrong when factoring
Don't write (x+3)(x+4) when you need numbers that add to -7 = wrong signs give +7 instead! The constant term 12 is positive but the middle term -7x is negative, so both factors must be negative. Always check: (-3) + (-4) = -7 and (-3) × (-4) = +12.

Practice Quiz

Test your knowledge with interactive questions

$$ x^2+6x+9=0 $$

What is the value of X?

FAQ

Everything you need to know about this question

How do I know which two numbers to pick for factoring?

Look for two numbers that multiply to give the constant term (12) and add to give the coefficient of x (-7). List factor pairs of 12: (1,12), (2,6), (3,4), then check which pair adds to -7.

What if I can't find factors that work?

If no integer factors work, the quadratic might not factor nicely. You can use the quadratic formula instead: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ .

Why do I get two different answers?

Quadratic equations usually have two solutions because a parabola crosses the x-axis at two points. Both x = 3 and x = 4 make the original equation equal zero.

How can I check my factoring is correct?

Expand your factored form using FOIL: (x-3)(x-4) = $x^2 - 4x - 3x + 12 = x^2 - 7x + 12$ . If it matches the original, you're right!

What does it mean when (x-3)(x-4) = 0?

This uses the Zero Product Property: if two things multiply to zero, at least one must be zero. So either x-3=0 (giving x=3) or x-4=0 (giving x=4).

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