Solving a System of Equations with Logarithms and Quadratics

Math Problem Statement

Solve the system of equations: log_x 8 = y + 1 and x^2 = 6 - x.

The system of equations in the image is as follows:

\log_x 8 = y + 1 \\ x^2 = 6 - x \end{cases}$$ ### Solving the system step by step: #### Step 1: Solve the second equation $$x^2 = 6 - x$$ This is a quadratic equation. Let's rearrange it: $$x^2 + x - 6 = 0$$ We can solve this using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ For this equation, $$a = 1$$, $$b = 1$$, and \(c =

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Logarithms
Quadratic Equations

Quadratic equation: ax^2 + bx + c = 0
Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
Logarithmic equation: log_b(a)

Quadratic formula
Properties of logarithms

Grades 9-11

Solve for y When x + y = 8 and log_2(xy) = 2

Solve for y in x + y = 8 and log_2(xy) = 2

Step-by-Step Solutions to Quadratic, Logarithmic, and Exponential Equations

Solve Simultaneous Logarithmic Equations ln(x) + ln(y^2) = 8 and ln(x^2) + ln(y) = 6

Solving Logarithmic System of Equations with Two Variables