Problem 48 - Algebra A Combined Function

Question

Solve. $$ \sqrt{x-2}+3=\sqrt{4 x+1} $$

Step-by-Step Solution

Answer

No real solutions satisfy the original equation.

Step 1: Isolate the square roots

First, we isolate one of the square roots. Let's isolate \( \sqrt{x-2} \) by subtracting 3 from both sides:\[\sqrt{x-2} = \sqrt{4x+1} - 3\]

Step 2: Square both sides

To eliminate the square roots, square both sides of the equation:\[(\sqrt{x-2})^2 = (\sqrt{4x+1} - 3)^2\]This simplifies to:\[x - 2 = (4x + 1) - 6\sqrt{4x + 1} + 9\]Simplify further:\[x - 2 = 4x + 10 - 6\sqrt{4x + 1}\]

Step 3: Rearrange terms

Move all terms except for the square root term to the left side:\[x - 4x - 10 = -6\sqrt{4x + 1} + 2\]Which simplifies to:\[-3x - 8 = -6\sqrt{4x + 1}\]

Step 4: Simplify and isolate the square root

Divide every term by -6 to isolate the square root:\[\frac{-3x - 8}{-6} = \sqrt{4x + 1}\]This simplifies to:\[\frac{3x + 8}{6} = \sqrt{4x + 1}\]

Step 5: Square both sides again

Square both sides again to remove the square root:\[\left(\frac{3x + 8}{6}\right)^2 = (\sqrt{4x + 1})^2\]Which simplifies to:\[\frac{(3x + 8)^2}{36} = 4x + 1\]

Step 6: Clear the fraction

Multiply every term by 36 to eliminate the fraction:\[(3x + 8)^2 = 36(4x + 1)\]

Step 7: Expand and solve the quadratic equation

Expand both sides:Left side:\[(3x + 8)^2 = 9x^2 + 48x + 64\]Right side:\[36(4x + 1) = 144x + 36\]Set the two expressions equal:\[9x^2 + 48x + 64 = 144x + 36\]

Step 8: Rearrange and solve for x

Rearrange to form a standard quadratic equation:\[9x^2 + 48x + 64 - 144x - 36 = 0\]Simplifying:\[9x^2 - 96x + 28 = 0\]Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 9\), \(b = -96\), and \(c = 28\).

Step 9: Apply the quadratic formula

Substitute into the quadratic formula:\[x = \frac{-(-96) \pm \sqrt{(-96)^2 - 4 \cdot 9 \cdot 28}}{2 \cdot 9}\]Calculate the discriminant and solve for \(x\):\[x = \frac{96 \pm \sqrt{9216 - 1008}}{18}\]Simplify:\[x = \frac{96 \pm \sqrt{8208}}{18}\]Continue solving to find possible values for \( x \).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool that helps solve quadratic equations of the form \( ax^2 + bx + c = 0 \). In this context, the quadratic equation arises when attempting to solve other types of equations, such as those involving square roots.
To use the quadratic formula:

• Identify the coefficients \( a \), \( b \), and \( c \) from the equation.
• The formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• The expression inside the square root, \( b^2 - 4ac \), is crucial and is called the discriminant.

The quadratic formula provides up to two solutions, which correspond to the values of \( x \) that make the quadratic equation true.

Square Root Isolation

Isolating the square root is a necessary step in solving equations involving square roots, as shown in the original exercise. Initially, you aim to have one of the square roots on one side of the equation.
For example, starting with \( \sqrt{x-2} + 3 = \sqrt{4x+1} \), we isolate \( \sqrt{x-2} \) by subtracting 3 from both sides, resulting in \( \sqrt{x-2} = \sqrt{4x+1} - 3 \).
This action prepares the expression for the next step, which typically involves squaring both sides to eliminate the root. This technique is essential in simplifying the problem to a form where other algebraic methods can be applied effectively.

Expanding Quadratics

Expanding quadratics is about expressing squared terms as a polynomial expression. It's a critical step when equations involve terms that are squared, such as \((3x + 8)^2\) seen in the problem.
When you expand \((3x + 8)^2\), you apply the formula \((a + b)^2 = a^2 + 2ab + b^2\):

• \((3x)^2 = 9x^2\)
• \(2 \times 3x \times 8 = 48x\)
• \(8^2 = 64\)

This results in \(9x^2 + 48x + 64\).
Expanding quadratics allows you to convert expressions involving squares into regular terms, simplifying the process of solving the quadratic equation later.

Discriminant Calculation

The discriminant is part of the quadratic formula, given by \( \Delta = b^2 - 4ac \). It informs you about the nature of the roots without the need to solve the entire equation immediately.
In the exercise, the discriminant for the equation \(9x^2 - 96x + 28 = 0\) is \((-96)^2 - 4 \times 9 \times 28\). Calculating this tells us:

• If \( \Delta > 0 \), there are two distinct real roots.
• If \( \Delta = 0 \), there's exactly one real root, meaning the roots are repeated (perfect square trinomial).
• If \( \Delta < 0 \), there are no real roots, and the equation has complex solutions.

By understanding the discriminant, you can predict the kind of solutions you'll get, which helps plan the next steps in problem-solving.