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

Q: Why do I need to divide by -1 first?

Dividing by -1 makes the leading coefficient positive, which is the standard form for factoring. It's much easier to factor $ x^2 + 7x + 12 $ than $ -x^2 - 7x - 12 $!

Q: How do I find the two numbers that work for factoring?

You need two numbers that multiply to 12 and add to 7. Try factor pairs of 12: (1,12), (2,6), (3,4). Since 3 + 4 = 7, use 3 and 4!

Q: Why are both solutions negative?

Look at the factored form: $ (x+3)(x+4) = 0 $. To make each factor zero, we need x = -3 and x = -4. Both solutions are negative because we're adding positive numbers to x to get zero.

Q: How do I check my answers?

Substitute each solution back into the original equation. For x = -3: $ -(-3)^2 - 7(-3) - 12 = -9 + 21 - 12 = 0 $ ✓

Quadratic Equations with Negative Leading Coefficients

Solve for x.

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 Multiply the entire exercise by minus to make it positive

00:15 Use trinomial to find the solution

00:19 Find 2 numbers whose product equals value C

00:23 And their sum equals value B

00:35 These are the matching numbers

00:45 Substitute these numbers in the trinomial

00:50 Find what zeros each factor

00:54 Isolate the unknown, this is one solution, now let's find the second

01:02 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

Solve for x.

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

Step-by-step solution

First, factor using trinomials and remember that there might be more than one solution for the value of $x$ :

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

Divide by -1:

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

$(x+3)(x+4)=0$

Therefore:

$x+4=0$

$x=-4$

Or:

$x+3=0$

$x=-3$

Final Answer

$x=-3,x=-4$

Key Points to Remember

Essential concepts to master this topic

Leading Coefficient: Divide by -1 to make the coefficient positive
Factoring: Find two numbers that multiply to 12 and add to 7
Check: Substitute x = -3: -(-3)² - 7(-3) - 12 = -9 + 21 - 12 = 0 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to divide the entire equation by -1
Don't just factor -x² - 7x - 12 directly = impossible to factor correctly! The negative leading coefficient makes factoring much harder. Always divide every term by -1 first to get x² + 7x + 12 = 0.

Practice Quiz

Test your knowledge with interactive questions

Find the value of the parameter x.

$$ 2x^2-7x+5=0 $$

FAQ

Everything you need to know about this question

Why do I need to divide by -1 first?

Dividing by -1 makes the leading coefficient positive, which is the standard form for factoring. It's much easier to factor $x^2 + 7x + 12$ than $-x^2 - 7x - 12$ !

How do I find the two numbers that work for factoring?

You need two numbers that multiply to 12 and add to 7. Try factor pairs of 12: (1,12), (2,6), (3,4). Since 3 + 4 = 7, use 3 and 4!

What if I can't factor the quadratic?

Not all quadratics can be factored easily! If factoring doesn't work, you can always use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$

Why are both solutions negative?

Look at the factored form: $(x+3)(x+4) = 0$ . To make each factor zero, we need x = -3 and x = -4. Both solutions are negative because we're adding positive numbers to x to get zero.

How do I check my answers?

Substitute each solution back into the original equation. For x = -3: $-(-3)^2 - 7(-3) - 12 = -9 + 21 - 12 = 0$ ✓

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