Problem 46 - Precalculus

Question

Solve the equations using any method you choose. (a) \(u(u+18)=-81\) (b) \(u(u+18)=81\)

Step-by-Step Solution

Answer

(a) Solutions: \(u = -9 \pm 9\sqrt{2}\); (b) Solutions: \(u = -9 \pm 9\sqrt{2}\).

Step 1: Rearrange Equation (a)

The equation given is \(u(u+18)=-81\). First, expand the equation to get: \(u^2 + 18u + 81 = 0\). This can be considered as a quadratic equation of the form \(au^2 + bu + c = 0\).

Step 2: Solve for u using the Quadratic Formula (a)

Use the quadratic formula \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve: Here, \(a = 1\), \(b = 18\), and \(c = 81\). Substitute these values into the quadratic formula: \(u = \frac{-18 \pm \sqrt{18^2 - 4 \times 1 \times (-81)}}{2 \times 1}\). Simplify to find the roots.

Step 3: Calculate Discriminant (a)

Calculate the discriminant: \(b^2 - 4ac = 18^2 - 4 \times 1 \times 81 = 324 + 324 = 648\).

Step 4: Find the Roots (a)

With the discriminant calculated, find the roots using: \(u = \frac{-18 \pm \sqrt{648}}{2}\).Simplify \(\sqrt{648} = 18\sqrt{2}\) so the expression becomes: \(u = \frac{-18 \pm 18\sqrt{2}}{2}\). This simplifies to: \(u_1 = -9 + 9\sqrt{2}\) and \(u_2 = -9 - 9\sqrt{2}\).

Step 5: Expand and Rearrange Equation (b)

The equation given is \(u(u+18)=81\). Expand the equation to get: \(u^2 + 18u - 81 = 0\). This is again a quadratic equation in the form \(au^2 + bu + c = 0\).

Step 6: Solve for u using the Quadratic Formula (b)

Use the quadratic formula \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve: Here, \(a = 1\), \(b = 18\), and \(c = -81\). Substitute these values into the quadratic formula: \(u = \frac{-18 \pm \sqrt{18^2 - 4 \times 1 \times (-81)}}{2 \times 1}\). Simplify to find the roots.

Step 7: Calculate Discriminant (b)

Calculate the discriminant: \(b^2 - 4ac = 18^2 + 4 \times 1 \times 81 = 324 + 324 = 648\).

Step 8: Find the Roots (b)

With the discriminant calculated as 648, find the roots using: \(u = \frac{-18 \pm \sqrt{648}}{2}\).Since \(\sqrt{648} = 18\sqrt{2}\), the expression becomes: \(u = \frac{-18 \pm 18\sqrt{2}}{2}\). This simplifies to: \(u_1 = -9 + 9\sqrt{2}\) and \(u_2 = -9 - 9\sqrt{2}\).

Key Concepts

quadratic formula

The quadratic formula is a powerful tool used to find the roots of quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). This formula can solve any quadratic equation, regardless of whether the roots are real or complex. The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]To use the formula, you need to identify the coefficients \(a\), \(b\), and \(c\) in your equation. Then, plug these coefficients into the formula to find the values of \(x\). The \(\pm\) symbol indicates that there can be two solutions: one involving addition and the other involving subtraction.
Here is a step-by-step guide:

• Identify the coefficients \(a\), \(b\), and \(c\) from the equation.
• Calculate the discriminant \( b^2 - 4ac \).
• Substitute \(a\), \(b\), and \(c\), plus the discriminant, into the quadratic formula.
• Simplify under the square root as much as possible before finalizing the roots.

This method ensures a comprehensive solution to any quadratic equation. It's especially useful when factoring is complex or impossible.

discriminant

The discriminant is a critical part of the quadratic formula, playing the key role in determining the nature of the roots of a quadratic equation.
The discriminant is given by the expression \(D = b^2 - 4ac\) formed from the coefficients of the quadratic equation. Here's why it's so important:

• **Positive Discriminant (\(D > 0\))**: The equation has two distinct real roots. This means that the quadratic intersects the x-axis at two distinct points.
• **Zero Discriminant (\(D = 0\))**: The equation has exactly one real root, or a ‘double root’. The parabola touches the x-axis at a single point, which is its vertex.
• **Negative Discriminant (\(D < 0\))**: The equation has two complex roots. In this case, there are no real solutions, and the parabola does not intersect the x-axis at all.

Calculating the discriminant is a useful step as it provides insight into what type of solutions to expect without fully solving the equation. This saves time and helps verify the correctness of the roots calculated.

expansion of equations

Expanding equations, specifically quadratics, involves taking a product of binomials and expressing it in the standard quadratic form \( ax^2 + bx + c = 0 \). This process is essential as it transforms a product into a sum, which then makes it possible to apply algebraic methods, like using the quadratic formula, to find solutions.
When given a quadratic of the form \( (x + p)(x + q) = r \), the expansion looks like this:

• Multiply the binomials together: \( (x + p)(x + q) = x^2 + qx + px + pq \).
• Simplify the expression: Combine like terms to get \( x^2 + (p+q)x + pq = r \).
• Rearrange the equation: Move all terms to one side, setting the equation to zero \( x^2 + (p+q)x + (pq-r) = 0 \).

By expanding, rearranging, and transforming the equation into this standard form, you're setting it up perfectly for solving with the quadratic formula. Understanding this process is crucial, especially when dealing with scenarios where equations come in factored form or involve constant terms on the other side of the equation. It ensures clarity and structure, two key components in solving quadratic equations efficiently.