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:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's find the value of X.

00:09 First, identify the coefficients in the equation.

00:25 Next, we use the quadratic formula. So, get ready.

00:47 Now, substitute the given values correctly and solve step by step.

01:06 Don't forget to calculate the squares and products separately.

01:22 Then, find the square root of sixteen. It's an important step.

01:30 Now, you'll have two possible solutions from addition and subtraction.

01:48 And that's how we solve this question! Great job!

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 $\sqrt{8} = 2\sqrt{2}$ .

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