Solve the Quadratic Equation: x²+10x+21=0 Step by Step

Q: Why are both roots negative in this problem?

When both b and c are positive in $ x^2 + bx + c = 0 $, the roots are always negative! This happens because we need two negative numbers that multiply to +21 and add to +10.

Q: Can I solve this by factoring instead?

Yes! Look for two numbers that multiply to 21 and add to 10. Since 3 × 7 = 21 and 3 + 7 = 10, we get $ (x + 3)(x + 7) = 0 $, so x = -3 or x = -7.

Q: What does the discriminant tell me?

The discriminant $ \Delta = b^2 - 4ac $ shows the nature of roots:Positive (like 16): Two different real rootsZero: One repeated rootNegative: No real roots

Q: How do I remember the quadratic formula?

Try this song: "x equals negative b, plus or minus the square root, of b squared minus 4ac, all over 2a!" Practice writing it out several times to build muscle memory.

Q: Why do I get the same answer with different methods?

That's exactly what should happen! Whether you use the quadratic formula, factoring, or completing the square, you'll always get the same roots. Different paths, same destination!

Q: What if my discriminant is not a perfect square?

No problem! You'll get irrational roots with square roots in them. For example, if $ \Delta = 8 $, your roots would contain $ \sqrt{8} = 2\sqrt{2} $.

Quadratic Equations with Factorization Approach

Solve the following equation:

$x^2+10x+21=0$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Identify the coefficients

00:19 Use the roots formula

00:41 Substitute appropriate values according to the given data and solve

01:00 Calculate the square and products

01:16 Calculate the square root of 16

01:24 These are the 2 possible solutions (addition,subtraction)

01:42 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 the following equation:

$x^2+10x+21=0$

Step-by-step solution

To solve this quadratic equation $x^2 + 10x + 21 = 0$ , we will use the quadratic formula. Follow these steps:

Identify the coefficients: $a = 1$ , $b = 10$ , $c = 21$ .
Calculate the discriminant $\Delta = b^2 - 4ac = 10^2 - 4 \cdot 1 \cdot 21 = 100 - 84 = 16$ .
The discriminant is positive, suggesting the equation has two distinct real roots.
Apply the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-10 \pm \sqrt{16}}{2 \cdot 1} = \frac{-10 \pm 4}{2}.$
Calculate the roots:
- First root: $x_1 = \frac{-10 + 4}{2} = \frac{-6}{2} = -3$ .
- Second root: $x_2 = \frac{-10 - 4}{2} = \frac{-14}{2} = -7$ .

Therefore, the solutions to the equation are $x_1 = -3$ and $x_2 = -7$ .

Comparing with the answer choices, the correct choice is $x_1 = -3, x_2 = -7$ .

Final Answer

$x_1=-3,x_2=-7$

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
Discriminant: $\Delta = 10^2 - 4(1)(21) = 100 - 84 = 16$
Verification: Check $(-3)^2 + 10(-3) + 21 = 9 - 30 + 21 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Sign errors when applying the quadratic formula
Don't forget the negative sign in front of b = making it +10 instead of -10! This flips all your signs and gives positive roots instead of negative ones. Always write $x = \frac{-b \pm \sqrt{\Delta}}{2a}$ with -b, not +b.

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 roots negative in this problem?

When both b and c are positive in $x^2 + bx + c = 0$ , the roots are always negative! This happens because we need two negative numbers that multiply to +21 and add to +10.

Can I solve this by factoring instead?

Yes! Look for two numbers that multiply to 21 and add to 10. Since 3 × 7 = 21 and 3 + 7 = 10, we get $(x + 3)(x + 7) = 0$ , so x = -3 or x = -7.

What does the discriminant tell me?

The discriminant $\Delta = b^2 - 4ac$ shows the nature of roots:

Positive (like 16): Two different real roots
Zero: One repeated root
Negative: No real roots

How do I remember the quadratic formula?

Try this song: "x equals negative b, plus or minus the square root, of b squared minus 4ac, all over 2a!" Practice writing it out several times to build muscle memory.

Why do I get the same answer with different methods?

That's exactly what should happen! Whether you use the quadratic formula, factoring, or completing the square, you'll always get the same roots. Different paths, same destination!

What if my discriminant is not a perfect square?

No problem! You'll get irrational roots with square roots in them. For example, if $\Delta = 8$ , your roots would contain $\sqrt{8} = 2\sqrt{2}$ .

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Solving Quadratic Equations questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime