Answer :
To solve the quadratic equation [tex]\( 2x^2 = 4x - 7 \)[/tex] for the values of [tex]\( x \)[/tex], we first need to rewrite it in standard form [tex]\( ax^2 + bx + c = 0 \)[/tex].
Rearranging the equation, we get:
[tex]\[ 2x^2 - 4x + 7 = 0 \][/tex]
Here, we identify the coefficients as:
[tex]\[ a = 2 \][/tex]
[tex]\[ b = -4 \][/tex]
[tex]\[ c = 7 \][/tex]
Next, we use the quadratic formula to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
First, we calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ \Delta = (-4)^2 - 4(2)(7) \][/tex]
[tex]\[ \Delta = 16 - 56 \][/tex]
[tex]\[ \Delta = -40 \][/tex]
The discriminant is [tex]\(-40\)[/tex], which is negative, indicating that the solutions will be complex numbers. Now, we substitute [tex]\(\Delta\)[/tex] back into the quadratic formula:
[tex]\[ x = \frac{-(-4) \pm \sqrt{-40}}{2(2)} \][/tex]
[tex]\[ x = \frac{4 \pm \sqrt{-40}}{4} \][/tex]
To simplify [tex]\(\sqrt{-40}\)[/tex], recall that:
[tex]\[ \sqrt{-40} = \sqrt{40} \cdot i = \sqrt{4 \cdot 10} \cdot i = 2\sqrt{10} \cdot i \][/tex]
Therefore:
[tex]\[ x = \frac{4 \pm 2\sqrt{10}i}{4} \][/tex]
Simplifying further:
[tex]\[ x = \frac{4}{4} \pm \frac{2\sqrt{10}i}{4} \][/tex]
[tex]\[ x = 1 \pm \frac{\sqrt{10}i}{2} \][/tex]
So, the solutions for [tex]\( x \)[/tex] are:
[tex]\[ x = 1 \pm \frac{\sqrt{10}i}{2} \][/tex]
The correct choice from the given options is:
[tex]\[ \frac{2 \pm \sqrt{10}i}{2} \][/tex]
Thus, the values of [tex]\( x \)[/tex] are [tex]\( \boxed{\frac{2 \pm \sqrt{10}i}{2}} \)[/tex].
Rearranging the equation, we get:
[tex]\[ 2x^2 - 4x + 7 = 0 \][/tex]
Here, we identify the coefficients as:
[tex]\[ a = 2 \][/tex]
[tex]\[ b = -4 \][/tex]
[tex]\[ c = 7 \][/tex]
Next, we use the quadratic formula to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
First, we calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ \Delta = (-4)^2 - 4(2)(7) \][/tex]
[tex]\[ \Delta = 16 - 56 \][/tex]
[tex]\[ \Delta = -40 \][/tex]
The discriminant is [tex]\(-40\)[/tex], which is negative, indicating that the solutions will be complex numbers. Now, we substitute [tex]\(\Delta\)[/tex] back into the quadratic formula:
[tex]\[ x = \frac{-(-4) \pm \sqrt{-40}}{2(2)} \][/tex]
[tex]\[ x = \frac{4 \pm \sqrt{-40}}{4} \][/tex]
To simplify [tex]\(\sqrt{-40}\)[/tex], recall that:
[tex]\[ \sqrt{-40} = \sqrt{40} \cdot i = \sqrt{4 \cdot 10} \cdot i = 2\sqrt{10} \cdot i \][/tex]
Therefore:
[tex]\[ x = \frac{4 \pm 2\sqrt{10}i}{4} \][/tex]
Simplifying further:
[tex]\[ x = \frac{4}{4} \pm \frac{2\sqrt{10}i}{4} \][/tex]
[tex]\[ x = 1 \pm \frac{\sqrt{10}i}{2} \][/tex]
So, the solutions for [tex]\( x \)[/tex] are:
[tex]\[ x = 1 \pm \frac{\sqrt{10}i}{2} \][/tex]
The correct choice from the given options is:
[tex]\[ \frac{2 \pm \sqrt{10}i}{2} \][/tex]
Thus, the values of [tex]\( x \)[/tex] are [tex]\( \boxed{\frac{2 \pm \sqrt{10}i}{2}} \)[/tex].
