Solve the Quadratic Equation: x²-19x+60=0 Step-by-Step

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

Look for factor pairs of the constant term (60). Try: 1×60, 2×30, 3×20, 4×15, 5×12, 6×10. Then check which pair adds up to the middle coefficient (19). Here, 4+15=19 works perfectly!

Q: Why do I get two different answers?

Quadratic equations typically have two solutions because when you multiply two expressions to get zero, either one could equal zero. That's why (x-4)(x-15)=0 gives both x=4 and x=15.

Q: How can I check if both answers are correct?

Substitute each answer back into the original equation. For x=4: $ 4^2-19(4)+60=16-76+60=0 $ ✓. For x=15: $ 15^2-19(15)+60=225-285+60=0 $ ✓

Q: What's the fastest way to factor quadratics?

Start by listing all factor pairs of the constant term, then quickly test which pair adds to the middle coefficient. With practice, you'll spot the right factors faster!

Quadratic Factoring with Integer Solutions

$x^2-19x+60=0$

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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 a trinomial, looking at coefficients

00:09 We want to find 2 numbers whose sum equals B (-19)

00:16 and their product equals C (60)

00:22 These are the matching numbers, we'll substitute in parentheses

00:31 We'll find what zeroes 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

$x^2-19x+60=0$

Step-by-step solution

Let's observe that the given equation:

$x^2-19x+60=0$ is a quadratic equation that can be solved using quick factoring:

$x^2-19x+60=0 \longleftrightarrow\begin{cases}\underline{?}\cdot\underline{?}=60\\ \underline{?}+\underline{?}=-19\end{cases}\\ \downarrow\\ (x-4)(x-15)=0$ and therefore we get two simpler equations from which we can extract the solution:

$(x-4)(x-15)=0 \\ \downarrow\\ x-4=0\rightarrow\boxed{x=4}\\ x-15=0\rightarrow\boxed{x=15}\\ \boxed{x=4,15}$ Therefore, the correct answer is answer A.

Final Answer

$x=15,x=4$

Key Points to Remember

Essential concepts to master this topic

Rule: Find two numbers that multiply to c and add to b
Technique: For $x^2-19x+60$ , find factors: 4×15=60 and 4+15=19
Check: Substitute x=4: $4^2-19(4)+60=0$ gives 16-76+60=0 ✓

Common Mistakes

Avoid these frequent errors

Using wrong signs when factoring negative middle terms
Don't write (x+4)(x+15)=0 when the middle term is -19x = wrong signs give positive 19x! The negative middle term means both factors must be negative: (x-4)(x-15). Always match the middle term's sign by making both factors negative when needed.

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 find the two numbers that work for factoring?

Look for factor pairs of the constant term (60). Try: 1×60, 2×30, 3×20, 4×15, 5×12, 6×10. Then check which pair adds up to the middle coefficient (19). Here, 4+15=19 works perfectly!

What if I can't find factors that work?

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

Why do I get two different answers?

Quadratic equations typically have two solutions because when you multiply two expressions to get zero, either one could equal zero. That's why (x-4)(x-15)=0 gives both x=4 and x=15.

How can I check if both answers are correct?

Substitute each answer back into the original equation. For x=4: $4^2-19(4)+60=16-76+60=0$ ✓. For x=15: $15^2-19(15)+60=225-285+60=0$ ✓

What's the fastest way to factor quadratics?

Start by listing all factor pairs of the constant term, then quickly test which pair adds to the middle coefficient. With practice, you'll spot the right factors faster!

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