Solve the Quadratic Equation: x² - 5x - 50 = 0 Step-by-Step

Q: How do I find the right factor pairs for -50?

List all factor pairs of 50: 1×50, 2×25, 5×10. Since you need a product of -50, one factor must be negative. Try different sign combinations until the sum equals the middle coefficient (-5).

Q: What if I can't factor this quadratic easily?

If factoring seems difficult, you can always use the quadratic formula: $ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} $. It works for any quadratic equation!

Q: Why do I get two answers from one equation?

Quadratic equations can have up to 2 solutions because they involve $ x^2 $. When $ (x-10)(x+5)=0 $, either factor can equal zero, giving you both solutions.

Q: Can I solve this using completing the square instead?

Yes! Completing the square always works, but factoring is usually faster when the numbers work out nicely. Save completing the square for when factoring gets messy.

Quadratic Factoring with Integer Coefficients

$x^2-5x-50=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's find X in this equation.

00:09 We'll factor the trinomial. Let's examine the coefficients closely.

00:14 We need two numbers that add up to B, which is five.

00:21 They should also multiply to give C, which is negative fifty.

00:29 We found the numbers! Let's put them in the parentheses.

00:36 Now, let's see what makes each factor equal zero.

00:46 And that's how we solve for X in this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$x^2-5x-50=0$

Step-by-step solution

Let's observe that the given equation:

$x^2-5x-50=0$ is a quadratic equation that can be solved using quick factoring:

$x^2-5x-50=0 \longleftrightarrow\begin{cases}\underline{?}\cdot\underline{?}=-50\\ \underline{?}+\underline{?}=-5\end{cases}\\ \downarrow\\ (x-10)(x+5)=0$ and therefore we get two simpler equations from which we can extract the solution:

$(x-10)(x+5)=0 \\ \downarrow\\ x-10=0\rightarrow\boxed{x=10}\\ x+5=0\rightarrow\boxed{x=-5}\\ \boxed{x=10,-5}$ Therefore, the correct answer is answer C.

Final Answer

$x=10,x=-5$

Key Points to Remember

Essential concepts to master this topic

Factoring Method: Find two numbers that multiply to -50 and add to -5
Technique: Use $(x-10)(x+5)=0$ since 10×(-5)=-50 and 10+(-5)=-5
Check: Substitute x=10: $10^2-5(10)-50=0$ ✓

Common Mistakes

Avoid these frequent errors

Using the wrong signs when factoring
Don't write (x+10)(x-5)=0 just because you see -50! This gives 10×(-5)=-50 but 10+(-5)=5, not -5. The middle term coefficient tells you the sum, so always check both the product AND sum of your factors.

Practice Quiz

Test your knowledge with interactive questions

Solve the following expression:

$$ x^2-1=0 $$

FAQ

Everything you need to know about this question

How do I find the right factor pairs for -50?

List all factor pairs of 50: 1×50, 2×25, 5×10. Since you need a product of -50, one factor must be negative. Try different sign combinations until the sum equals the middle coefficient (-5).

What if I can't factor this quadratic easily?

If factoring seems difficult, you can always use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ . It works for any quadratic equation!

Why do I get two answers from one equation?

Quadratic equations can have up to 2 solutions because they involve $x^2$ . When $(x-10)(x+5)=0$ , either factor can equal zero, giving you both solutions.

How do I know which factor pair to choose?

Test each pair! For -50, try: 1×(-50)=-50 but 1+(-50)=-49 ✗. Try 10×(-5)=-50 and 10+(-5)=5 ✗. Try (-10)×5=-50 and (-10)+5=-5 ✓

Can I solve this using completing the square instead?

Yes! Completing the square always works, but factoring is usually faster when the numbers work out nicely. Save completing the square for when factoring gets messy.

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