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Find all the roots and check the answer.

[tex]\[
\begin{array}{l}
3x^2 + 22x + 7 = 0 \\
x = \square
\end{array}
\][/tex]

(Use a comma to separate answers as needed. Type an integer or a simplified fraction.)


Sagot :

To solve the quadratic equation [tex]\( 3x^2 + 22x + 7 = 0 \)[/tex], we can use the quadratic formula:

[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

For the given equation, [tex]\(a = 3\)[/tex], [tex]\(b = 22\)[/tex], and [tex]\(c = 7\)[/tex]. Let's break down the solution step-by-step.

1. Calculate the Discriminant:

[tex]\[ \text{Discriminant} = b^2 - 4ac \][/tex]

[tex]\[ \text{Discriminant} = 22^2 - 4 \times 3 \times 7 \][/tex]

[tex]\[ \text{Discriminant} = 484 - 84 = 400 \][/tex]

2. Calculate the Roots:

Now, substitute the values into the quadratic formula:

[tex]\[ x = \frac{-b \pm \sqrt{\text{Discriminant}}}{2a} \][/tex]

[tex]\[ x = \frac{-22 \pm \sqrt{400}}{2 \times 3} \][/tex]

[tex]\[ x = \frac{-22 \pm 20}{6} \][/tex]

We have two cases to consider:

[tex]\[ x_1 = \frac{-22 + 20}{6} = \frac{-2}{6} = -\frac{1}{3} \][/tex]

[tex]\[ x_2 = \frac{-22 - 20}{6} = \frac{-42}{6} = -7 \][/tex]

So, the roots of the equation are:

[tex]\[ x = -\frac{1}{3}, -7 \][/tex]

3. Check the Roots:

We can verify the roots by substituting them back into the original equation:

For [tex]\( x = -\frac{1}{3} \)[/tex]:

[tex]\[ 3\left(-\frac{1}{3}\right)^2 + 22\left(-\frac{1}{3}\right) + 7 \][/tex]

[tex]\[ 3 \times \frac{1}{9} - \frac{22}{3} + 7 \][/tex]

[tex]\[ \frac{3}{9} - \frac{22}{3} + 7 = \frac{1}{3} - \frac{22}{3} + 7 \][/tex]

[tex]\[ = \frac{1 - 22 + 21}{3} = \frac{0}{3} = 0 \][/tex]

For [tex]\( x = -7 \)[/tex]:

[tex]\[ 3(-7)^2 + 22(-7) + 7 \][/tex]

[tex]\[ 3 \times 49 - 154 + 7 \][/tex]

[tex]\[ 147 - 154 + 7 = 0 \][/tex]

Both roots satisfy the original equation, confirming that the solutions are correct.

Thus, the solutions to the equation [tex]\( 3x^2 + 22x + 7 = 0 \)[/tex] are:

[tex]\[ \boxed{-\frac{1}{3}, -7} \][/tex]
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