Solve X²+10X+9=0: Complete Quadratic Equation Guide

Q: Why are both solutions negative in this problem?

When you have $ X^2+10X+9=0 $, the positive coefficient of X (which is +10) combined with the positive constant term creates a situation where both roots are negative. This happens because we need two negative numbers that multiply to +9 and add to -10.

Q: How do I remember the quadratic formula correctly?

Remember: "negative b, plus or minus..." The formula always starts with -b, not +b. Practice writing it as $ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} $ several times!

Q: Can I solve this by factoring instead?

Yes! You can factor $ X^2+10X+9 $ as $ (X+1)(X+9)=0 $. This gives X+1=0 or X+9=0, so X=-1 or X=-9. Both methods give the same answer!

Q: What does the discriminant tell me?

The discriminant is $ b^2-4ac = 100-36 = 64 $. Since it's positive and a perfect square, you get two different rational solutions. If it were negative, there would be no real solutions.

Q: How can I check if my solutions are correct?

Substitute each solution back into the original equation:For X=-1: $ (-1)^2 + 10(-1) + 9 = 1 - 10 + 9 = 0 $ ✓For X=-9: $ (-9)^2 + 10(-9) + 9 = 81 - 90 + 9 = 0 $ ✓

Quadratic Formula with Negative Solutions

What is the value of X in the following equation?

$X^2+10X+9=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Let's pay attention to the coefficients

00:14 We want to find 2 numbers whose sum equals B(10)

00:17 and their product equals C(9)

00:24 These are the matching numbers

00:28 Let's substitute these solutions in the multiplication equation

00:35 Let's find what makes each factor zero

00:44 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

What is the value of X in the following equation?

$X^2+10X+9=0$

Step-by-step solution

To answer the question, we'll need to recall the quadratic formula:

$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$

Let's remember that:

a is the coefficient of X²

b is the coefficient of X

c is the free term

And if we look again at the formula given to us:

a=1

b=10

c=9

Let's substitute into the formula:

$x = {-10 \pm \sqrt{10^2-4\cdot 1 \cdot 9} \over 2\cdot 1}$

Let's start by solving what's under the square root:

$x = {-10 \pm \sqrt{100-36} \over 2}$

$x = {-10 \pm \sqrt{64} \over 2}$

$x = {-10 \pm 8 \over 2}$

Now we'll solve twice, once with plus and once with minus

$x = {-10 +8 \over 2}= {-2 \over 2} = -1$

$x = {-10 -8 \over 2} = {-18 \over 2} =-9$

And we can see that we got two solutions, X=-1 and X=-9

And that's the solution!

Final Answer

$x_1=-1,x_2=-9$

Key Points to Remember

Essential concepts to master this topic

Formula: Use $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ for any quadratic equation
Technique: Substitute a=1, b=10, c=9 to get $x = \frac{-10 \pm 8}{2}$
Check: Both solutions: (-1)²+10(-1)+9 = 0 and (-9)²+10(-9)+9 = 0 ✓

Common Mistakes

Avoid these frequent errors

Getting positive solutions instead of negative ones
Don't forget the negative sign in front of b in the quadratic formula = x₁=1, x₂=9 instead of correct x₁=-1, x₂=-9! The formula starts with -b, not +b. Always write -10 when b=10, giving you both negative solutions.

Practice Quiz

Test your knowledge with interactive questions

a = Coefficient of x²

b = Coefficient of x

c = Coefficient of the independent number

what is the value of $$ a $$ in the equation

$$ y=3x-10+5x^2 $$

FAQ

Everything you need to know about this question

Why are both solutions negative in this problem?

When you have $X^2+10X+9=0$ , the positive coefficient of X (which is +10) combined with the positive constant term creates a situation where both roots are negative. This happens because we need two negative numbers that multiply to +9 and add to -10.

How do I remember the quadratic formula correctly?

Remember: "negative b, plus or minus..." The formula always starts with -b, not +b. Practice writing it as $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ several times!

Can I solve this by factoring instead?

Yes! You can factor $X^2+10X+9$ as $(X+1)(X+9)=0$ . This gives X+1=0 or X+9=0, so X=-1 or X=-9. Both methods give the same answer!

What does the discriminant tell me?

The discriminant is $b^2-4ac = 100-36 = 64$ . Since it's positive and a perfect square, you get two different rational solutions. If it were negative, there would be no real solutions.

How can I check if my solutions are correct?

Substitute each solution back into the original equation:

For X=-1: $(-1)^2 + 10(-1) + 9 = 1 - 10 + 9 = 0$ ✓
For X=-9: $(-9)^2 + 10(-9) + 9 = 81 - 90 + 9 = 0$ ✓

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