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Given [tex]\(4x^2 - 7x = -3\)[/tex], what are the solutions?

A. [tex]\(x = \left\{\frac{3}{4}, 1\right\}\)[/tex]

B. [tex]\(x = \left\{\frac{7 \pm \sqrt{97}}{8}\right\}\)[/tex]

C. [tex]\(x = \left\{-\frac{3}{4}, -1\right\}\)[/tex]

D. [tex]\(x = \left\{\frac{-7 \pm \sqrt{97}}{8}\right\}\)[/tex]


Sagot :

To solve the quadratic equation [tex]\(4x^2 - 7x = -3\)[/tex], let's first rearrange it into standard form:

[tex]\[ 4x^2 - 7x + 3 = 0 \][/tex]

Next, we’ll use the quadratic formula to find the roots of the equation [tex]\(ax^2 + bx + c = 0\)[/tex]. The quadratic formula is given by:

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

For the given equation [tex]\(4x^2 - 7x + 3 = 0\)[/tex], the coefficients are:
- [tex]\(a = 4\)[/tex]
- [tex]\(b = -7\)[/tex]
- [tex]\(c = 3\)[/tex]

Now, let's substitute these values into the quadratic formula:

[tex]\[ x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4 \cdot 4 \cdot 3}}{2 \cdot 4} \][/tex]

First, calculate the discriminant ([tex]\(b^2 - 4ac\)[/tex]):

[tex]\[ b^2 - 4ac = (-7)^2 - 4 \cdot 4 \cdot 3 = 49 - 48 = 1 \][/tex]

Now, substitute the discriminant back into the quadratic formula:

[tex]\[ x = \frac{7 \pm \sqrt{1}}{8} \][/tex]

Since [tex]\(\sqrt{1} = 1\)[/tex], we can simplify this further:

[tex]\[ x = \frac{7 \pm 1}{8} \][/tex]

We now have two potential solutions:
[tex]\[ x_1 = \frac{7 + 1}{8} = \frac{8}{8} = 1 \][/tex]
[tex]\[ x_2 = \frac{7 - 1}{8} = \frac{6}{8} = \frac{3}{4} \][/tex]

Thus, the solutions to the equation [tex]\(4x^2 - 7x + 3 = 0\)[/tex] are:

[tex]\[ x = \left\{ \frac{3}{4}, 1 \right\} \][/tex]

So, the correct answer is:

A. [tex]\(x = \left\{ \frac{3}{4}, 1 \right\} \)[/tex]