Solve for [tex]\( x \)[/tex]:
[tex]\[ 3x = 6x - 2 \][/tex]


Format the following question or task so that it is easier to read. Remove all unnecessary portions of the text.
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[tex]\[ 2x^2 + 7x - 4 = 0 \][/tex]

Identify the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:

[tex]\[ a = \_\_\_ \][/tex]
[tex]\[ b = \_\_\_ \][/tex]
[tex]\[ c = \_\_\_ \][/tex]



Answer :

Certainly! Let's solve the quadratic equation [tex]\(2x^2 + 7x - 4 = 0\)[/tex] step by step.

1. Identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
The quadratic equation is given in standard form [tex]\(ax^2 + bx + c = 0\)[/tex].
- [tex]\(a = 2\)[/tex]
- [tex]\(b = 7\)[/tex]
- [tex]\(c = -4\)[/tex]

2. Write down the quadratic formula:
The quadratic formula for finding the roots of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

3. Calculate the discriminant:
The discriminant is the part of the quadratic formula under the square root:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = 7^2 - 4(2)(-4) = 49 + 32 = 81 \][/tex]

4. Calculate the two roots:
Using the discriminant, we now find the two possible values for [tex]\(x\)[/tex] using the plus and minus in the quadratic formula.

- First root:
[tex]\[ x_1 = \frac{-b + \sqrt{\Delta}}{2a} = \frac{-7 + \sqrt{81}}{2 \cdot 2} = \frac{-7 + 9}{4} = \frac{2}{4} = 0.5 \][/tex]

- Second root:
[tex]\[ x_2 = \frac{-b - \sqrt{\Delta}}{2a} = \frac{-7 - \sqrt{81}}{2 \cdot 2} = \frac{-7 - 9}{4} = \frac{-16}{4} = -4 \][/tex]

5. Summary:
- The discriminant [tex]\(\Delta\)[/tex] is 81.
- The first root [tex]\(x_1\)[/tex] is 0.5.
- The second root [tex]\(x_2\)[/tex] is -4.0.

So the solutions of the quadratic equation [tex]\(2x^2 + 7x - 4 = 0\)[/tex] are:
[tex]\[ x_1 = 0.5 \quad \text{and} \quad x_2 = -4 \][/tex]