Question

If \(k=\frac{x+3}{x-4}\) and \(k^{2}-3 k=28,\) find \(x\)

Short answer

The values of \(x\) are 2.6 and 5.1667.

Step-by-step solution

Rewrite the initial equation

Start with the given equation: \[k = \frac{x+3}{x-4}\]This means that we can substitute \[\frac{x+3}{x-4}\] for \(k\) in the second equation.

Substitute the value of k

The second given equation is:\[k^{2} - 3k = 28\]Substitute \(\frac{x+3}{x-4}\) for \(k\):\[\left(\frac{x+3}{x-4}\right)^{2} - 3\left(\frac{x+3}{x-4}\right) = 28\]

Simplify the equation

To simplify, clear the fraction by multiplying every term by \((x-4)^{2}\):\[(x+3)^{2} - 3(x+3)(x-4) = 28(x-4)^{2}\]

Expand both sides

Expand each term:On the left side,\[(x+3)^{2} = x^{2} + 6x + 9\]and\[-3(x+3)(x-4) = -3(x^{2} - x - 12) = -3x^{2} + 3x + 36\]And on the right side,\[28(x-4)^{2} = 28(x^{2} - 8x + 16) = 28x^{2} - 224x + 448\]

Combine like terms

Combine all terms on one side of the equation:\[x^{2} + 6x + 9 - 3x^{2} + 3x + 36 = 28x^{2} - 224 + 448\]This results in:\[-2x^{2} + 9x + 45 = 28x^{2} - 224x + 448\]

Simplify into a standard quadratic form

Move all terms to one side of the equation to set it to zero:\[-2x^{2} + 9x + 45 - 28x^{2} + 224x - 448 = 0\]which simplifies to:\[ -30x^{2} + 233x - 403 = 0 \]

Solve the quadratic equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) with the equation \(-30x^{2} + 233x - 403 = 0\).Identify the coefficients: \(a = -30\), \(b = 233\), and \(c = -403\).Substitute these into the quadratic formula:\[x = \frac{-233 \pm \sqrt{233^2 - 4(-30)(-403)}}{2(-30)}\]

Calculate the discriminant and solve

First, calculate the discriminant:\[233^2 - 4(-30)(-403) = 54289 - 48360 = 5929\]Then compute:\[x = \frac{-233 \pm \sqrt{5929}}{-60}\]This simplifies to:\[x = \frac{-233 \pm 77}{-60}\]

Find the roots

Calculate the two possible values for \(x\):\[x = \frac{-233 + 77}{-60} = \frac{-156}{-60} = 2.6\]and\[x = \frac{-233 - 77}{-60} = \frac{-310}{-60} = 5.1667\]

Key concepts

quadratic equations

A quadratic equation is any equation that can be written in the form \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are constants. In the problem we are solving, we ended up with a quadratic equation, which is \( -30x^2 + 233x - 403 = 0 \). Quadratic equations can have at most two solutions. To solve them, we often use the quadratic formula: \( x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \). In this formula,\(a\) is the coefficient of \(x^2\), \(b\) the coefficient of \(x\), and \(c\) the constant term. This formula helps us find the values of \(x\) that satisfy the equation.

substitution method

The substitution method is a technique used to solve systems of equations or simplify complex equations. In our exercise, we had two expressions involving \(k\) and \(x\). One of them was \( k = \frac{x+3}{x-4} \), and the other was \( k^2 - 3k = 28 \). By substituting \( \frac{x+3}{x-4} \) in place of \(k\) in the second equation, we get \( \left( \frac{x+3}{x-4} \right)^{2} - 3 \left( \frac{x+3}{x-4} \right) = 28 \). This substitution makes it possible to solve the equation in terms of \(x\).

discriminant

The discriminant is a key concept in solving quadratic equations. It’s found in the quadratic formula and calculated as \( b^2 - 4ac \). The value of the discriminant tells us about the nature of the roots of the quadratic equation:

• If the discriminant is positive (>0), there are two distinct real roots.
• If the discriminant is zero (=0), there is exactly one real root (a repeated root).
• If the discriminant is negative (<0), there are no real roots, but instead two complex conjugate roots.

In our example, the discriminant was \( b^2 - 4ac = 233^2 - 4(-30)(-403) = 5929 \). Since it was positive, we knew the equation had two distinct real roots. These roots were calculated as \( x = \frac{-233 \, \pm \, \sqrt{5929}}{-60} \), which gave us the solutions \( x = 2.6 \) and \( x = 5.1667 \).