Represent the Product of the Quadratic Function: x²-5x-50

Q: Why do we need two numbers that both multiply AND add to specific values?

This comes from the FOIL method in reverse! When you expand $ (x+a)(x+b) $, you get $ x^2 + (a+b)x + ab $. So the middle term coefficient equals a + b and the constant term equals a × b.

Q: What if I can't find factor pairs that work?

List all possible factor pairs of the constant term systematically. For -50: (±1, ±50), (±2, ±25), (±5, ±10). Check each pair's sum until you find the right one!

Q: How do I know which signs to use in the factors?

Look at the signs in your quadratic! If the constant is negative (like -50), your factors must have opposite signs. The larger absolute value gets the sign that matches the middle term's sign.

Q: Can I check my answer without expanding?

Yes! Find where each factor equals zero: $ x+5=0 $ gives $ x=-5 $, and $ x-10=0 $ gives $ x=10 $. Substitute these into the original equation - both should equal zero.

Q: What if the leading coefficient isn't 1?

This method works best when the coefficient of $ x^2 $ is 1. If it's not, you might need different techniques like grouping or the quadratic formula for more complex factoring.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Convert the trinom to multiplication

00:03 Identify the coefficients of the trinom

00:07 We want to find 2 numbers whose sum equals coefficient B

00:11 and their product equals coefficient C

00:18 These are the matching numbers, let's substitute in the multiplication

00:25 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Find the representation of the product of the following function

$f(x)=x^2-5x-50$

2

Step-by-step solution

To solve this problem, we will factor the quadratic function $f(x) = x^2 - 5x - 50$ into two binomials:

Step 1: Identify the values of $a = 1$ , $b = -5$ , and $c = -50$ in the quadratic expression $ax^2 + bx + c$ .
Step 2: We look for two numbers that multiply to $c = -50$ and add up to $b = -5$ .
Step 3: Consider the factor pairs of $-50$ . Possible pairs include $(-1, 50)$ , $(1, -50)$ , $(-2, 25)$ , $(2, -25)$ , $(-5, 10)$ , and $(5, -10)$ .
Step 4: The correct pair is $(5, -10)$ because $5 \times (-10) = -50$ and $5 + (-10) = -5$ .
Step 5: Express $f(x)$ in its factored form using these numbers: $f(x) = (x + 5)(x - 10)$ .
Step 6: Verify the factorization by expanding: $(x + 5)(x - 10) = x^2 - 10x + 5x - 50$ , which simplifies to $x^2 - 5x - 50$ , confirming correctness.

Therefore, the correct factorization of the quadratic $f(x) = x^2 - 5x - 50$ is $(x + 5)(x - 10)$ .

Thus, the product representation of the function is $(x+5)(x-10)$ .

3

Final Answer

$(x+5)(x-10)$

Key Points to Remember

Essential concepts to master this topic

Rule: Find two numbers that multiply to c and add to b
Technique: Factor pairs of -50: (5, -10) since 5 × (-10) = -50 and 5 + (-10) = -5
Check: Expand $(x+5)(x-10) = x^2-5x-50$ to verify factorization ✓

Common Mistakes

Avoid these frequent errors

Confusing addition and multiplication requirements for factor pairs
Don't just find numbers that multiply to -50 without checking if they add to -5 = wrong factorization like (x-1)(x+50)! You need BOTH conditions: multiply to c AND add to b coefficient. Always verify both multiplication and addition before writing your final answer.

Practice Quiz

Test your knowledge with interactive questions

Create an algebraic expression based on the following parameters:

$$ a=3,b=0,c=-3 $$

FAQ

Everything you need to know about this question

Why do we need two numbers that both multiply AND add to specific values?

+

This comes from the FOIL method in reverse! When you expand $(x+a)(x+b)$ , you get $x^2 + (a+b)x + ab$ . So the middle term coefficient equals a + b and the constant term equals a × b.

What if I can't find factor pairs that work?

+

List all possible factor pairs of the constant term systematically. For -50: (±1, ±50), (±2, ±25), (±5, ±10). Check each pair's sum until you find the right one!

How do I know which signs to use in the factors?

+

Look at the signs in your quadratic! If the constant is negative (like -50), your factors must have opposite signs. The larger absolute value gets the sign that matches the middle term's sign.

Can I check my answer without expanding?

+

Yes! Find where each factor equals zero: $x+5=0$ gives $x=-5$ , and $x-10=0$ gives $x=10$ . Substitute these into the original equation - both should equal zero.

What if the leading coefficient isn't 1?

+

This method works best when the coefficient of $x^2$ is 1. If it's not, you might need different techniques like grouping or the quadratic formula for more complex factoring.

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Represent the Product of the Quadratic Function: x²-5x-50

Quadratic Functions with Factoring Method

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Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do we need two numbers that both multiply AND add to specific values?

What if I can't find factor pairs that work?

How do I know which signs to use in the factors?

Can I check my answer without expanding?

What if the leading coefficient isn't 1?

🌟 Unlock Your Math Potential

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