What are the solutions to the equation log_{10}(4)(x^2-x+2)= 1/2 ?

The solutions to the equation log_{10}(4)(x^2-x+2)= 1/2 are

Find the zeros of log_{10}(4)(x^2-x+2)= 1/2

The zeros of log_{10}(4)(x^2-x+2)= 1/2 are

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lo g_{10} (4) (x^{2} - x + 2) = \frac{1}{2}

x = \frac{1}{2} + i \frac{112 lo g _{10} ( 2 ) ^{2} - 16 lo g _{10} ( 2 )}{8 lo g _{10} ( 2 )}, x = \frac{1}{2} - i \frac{112 lo g _{10} ( 2 ) ^{2} - 16 lo g _{10} ( 2 )}{8 lo g _{10} ( 2 )}

Solution steps

lo g_{10} (4) (x^{2} - x + 2) = \frac{1}{2}

Multiply both sides by

2

4 lo g_{10} (2) (x^{2} - x + 2) = 1

Expand

4 lo g_{10} (2) (x^{2} - x + 2) : 4 lo g_{10} (2) x^{2} - 4 lo g_{10} (2) x + 8 lo g_{10} (2)

4 lo g_{10} (2) x^{2} - 4 lo g_{10} (2) x + 8 lo g_{10} (2) = 1

Move

1

to the left side

4 lo g_{10} (2) x^{2} - 4 lo g_{10} (2) x + 8 lo g_{10} (2) - 1 = 0

Solve with the quadratic formula

x_{1, 2} = \frac{- ( - 4 lo g _{10} ( 2 ) ) \pm ( - 4 lo g _{10} ( 2 ) ) ^{2} - 4 \cdot 4 lo g _{10} ( 2 ) ( 8 lo g _{10} ( 2 ) - 1 )}{2 \cdot 4 lo g _{10} ( 2 )}

Simplify

(- 4 lo g_{10} (2))^{2} - 4 \cdot 4 lo g_{10} (2) (8 lo g_{10} (2) - 1) : i 112 lo g_{10} (2)^{2} - 16 lo g_{10} (2)

x_{1, 2} = \frac{- ( - 4 lo g _{10} ( 2 ) ) \pm i 112 lo g _{10} ( 2 ) ^{2} - 16 lo g _{10} ( 2 )}{2 \cdot 4 lo g _{10} ( 2 )}

Separate the solutions

x_{1} = \frac{- ( - 4 lo g _{10} ( 2 ) ) + i 112 lo g _{10} ( 2 ) ^{2} - 16 lo g _{10} ( 2 )}{2 \cdot 4 lo g _{10} ( 2 )}, x_{2} = \frac{- ( - 4 lo g _{10} ( 2 ) ) - i 112 lo g _{10} ( 2 ) ^{2} - 16 lo g _{10} ( 2 )}{2 \cdot 4 lo g _{10} ( 2 )}

x = \frac{- ( - 4 lo g _{10} ( 2 ) ) + i 112 lo g _{10} ( 2 ) ^{2} - 16 lo g _{10} ( 2 )}{2 \cdot 4 lo g _{10} ( 2 )} : \frac{1}{2} + i \frac{112 lo g _{10} ( 2 ) ^{2} - 16 lo g _{10} ( 2 )}{8 lo g _{10} ( 2 )}

x = \frac{- ( - 4 lo g _{10} ( 2 ) ) - i 112 lo g _{10} ( 2 ) ^{2} - 16 lo g _{10} ( 2 )}{2 \cdot 4 lo g _{10} ( 2 )} : \frac{1}{2} - i \frac{112 lo g _{10} ( 2 ) ^{2} - 16 lo g _{10} ( 2 )}{8 lo g _{10} ( 2 )}

The solutions to the quadratic equation are:

x = \frac{1}{2} + i \frac{112 lo g _{10} ( 2 ) ^{2} - 16 lo g _{10} ( 2 )}{8 lo g _{10} ( 2 )}, x = \frac{1}{2} - i \frac{112 lo g _{10} ( 2 ) ^{2} - 16 lo g _{10} ( 2 )}{8 lo g _{10} ( 2 )}

3 x^{2} + 180 = - 48 x

2 x^{2} = - 20

x^{2} - 4 x

x^{2} + 6 x + 10

x^{2} + 6 x + 25

What are the solutions to the equation log_{10}(4)(x^2-x+2)= 1/2 ?
The solutions to the equation log_{10}(4)(x^2-x+2)= 1/2 are
Find the zeros of log_{10}(4)(x^2-x+2)= 1/2
The zeros of log_{10}(4)(x^2-x+2)= 1/2 are