Question
Quauratic Equations Quick Check Solve the quadratic equation 10x^2-2x+13=3. Which of the following expresses its solutions in the form a± bi ? (1 point) 1,3sqrt(11).
Answers
Answered by
GPT-4o mini
To solve the quadratic equation \(10x^2 - 2x + 13 = 3\), we first rearrange it into standard form:
\[
10x^2 - 2x + 13 - 3 = 0
\]
\[
10x^2 - 2x + 10 = 0
\]
Now we can apply the quadratic formula, which is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, in our equation \(10x^2 - 2x + 10 = 0\), we have:
- \(a = 10\)
- \(b = -2\)
- \(c = 10\)
Substituting these values into the quadratic formula:
1. Calculate the discriminant \(b^2 - 4ac\):
\[
b^2 - 4ac = (-2)^2 - 4(10)(10) = 4 - 400 = -396
\]
2. Since the discriminant is negative, the solutions will be complex.
Now apply the quadratic formula:
\[
x = \frac{-(-2) \pm \sqrt{-396}}{2(10)} = \frac{2 \pm \sqrt{-396}}{20}
\]
3. Simplify \(\sqrt{-396}\):
\[
\sqrt{-396} = \sqrt{396} i = \sqrt{4 \cdot 99} i = 2\sqrt{99} i = 2\sqrt{9 \cdot 11} i = 6\sqrt{11} i
\]
4. Substitute back into the quadratic formula:
\[
x = \frac{2 \pm 6\sqrt{11} i}{20} = \frac{2}{20} \pm \frac{6\sqrt{11} i}{20}
\]
5. Simplify:
\[
x = \frac{1}{10} \pm \frac{3\sqrt{11}}{10} i
\]
Thus, the solutions of the quadratic equation \(10x^2 - 2x + 10 = 0\) expressed in the form \(a ± bi\) are:
\[
\frac{1}{10} \pm \frac{3\sqrt{11}}{10} i
\]
So the answer is:
\[
\frac{1}{10} \pm \frac{3\sqrt{11}}{10} i
\]
\[
10x^2 - 2x + 13 - 3 = 0
\]
\[
10x^2 - 2x + 10 = 0
\]
Now we can apply the quadratic formula, which is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, in our equation \(10x^2 - 2x + 10 = 0\), we have:
- \(a = 10\)
- \(b = -2\)
- \(c = 10\)
Substituting these values into the quadratic formula:
1. Calculate the discriminant \(b^2 - 4ac\):
\[
b^2 - 4ac = (-2)^2 - 4(10)(10) = 4 - 400 = -396
\]
2. Since the discriminant is negative, the solutions will be complex.
Now apply the quadratic formula:
\[
x = \frac{-(-2) \pm \sqrt{-396}}{2(10)} = \frac{2 \pm \sqrt{-396}}{20}
\]
3. Simplify \(\sqrt{-396}\):
\[
\sqrt{-396} = \sqrt{396} i = \sqrt{4 \cdot 99} i = 2\sqrt{99} i = 2\sqrt{9 \cdot 11} i = 6\sqrt{11} i
\]
4. Substitute back into the quadratic formula:
\[
x = \frac{2 \pm 6\sqrt{11} i}{20} = \frac{2}{20} \pm \frac{6\sqrt{11} i}{20}
\]
5. Simplify:
\[
x = \frac{1}{10} \pm \frac{3\sqrt{11}}{10} i
\]
Thus, the solutions of the quadratic equation \(10x^2 - 2x + 10 = 0\) expressed in the form \(a ± bi\) are:
\[
\frac{1}{10} \pm \frac{3\sqrt{11}}{10} i
\]
So the answer is:
\[
\frac{1}{10} \pm \frac{3\sqrt{11}}{10} i
\]