To solve the quadratic equation [tex]\( 4x^2 = 9x - 3 \)[/tex], we'll follow these steps:
1. Rewrite the equation in standard form:
The standard form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex].
We start with:
[tex]\[
4x^2 = 9x - 3
\][/tex]
To convert this into the standard form, move all terms to one side of the equation:
[tex]\[
4x^2 - 9x + 3 = 0
\][/tex]
Now, we can identify the coefficients:
[tex]\[
a = 4, \quad b = -9, \quad c = 3
\][/tex]
2. Calculate the discriminant:
The discriminant of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\Delta = (-9)^2 - 4 \cdot 4 \cdot 3 = 81 - 48 = 33
\][/tex]
3. Use the quadratic formula:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and the discriminant [tex]\( \Delta \)[/tex]:
[tex]\[
x = \frac{9 \pm \sqrt{33}}{8}
\][/tex]
4. Calculate the two solutions:
We compute the two potential solutions for [tex]\( x \)[/tex]:
[tex]\[
x_1 = \frac{9 + \sqrt{33}}{8}
\][/tex]
[tex]\[
x_2 = \frac{9 - \sqrt{33}}{8}
\][/tex]
5. Round the solutions to the nearest hundredth:
Finally, we round the solutions to the nearest hundredth. Given:
[tex]\[
x_1 \approx 1.84 \quad \text{and} \quad x_2 \approx 0.41
\][/tex]
Therefore, the solutions to the quadratic equation [tex]\( 4x^2 = 9x - 3 \)[/tex] rounded to the nearest hundredth are:
[tex]\[
x = 1.84, 0.41
\][/tex]
