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Chapter 10: Problem 68

Solve the quadratic equation by completing the square. $$x^{2}-9 x+16=0$$

Short Answer

Expert verified
The solutions to the equation are \(x=4.5+\sqrt{4.25}\) and \(x=4.5-\sqrt{4.25}\).

Step by step solution

01

Rewriting Equation

The equation looks like this: \(x^{2}-9 x+16=0\). It can be rewritten as \(x^{2}-9x=-16\).
02

Complete the Square

Half of coefficient of x is \(-\frac{9}{2}\), squaring it results in \(20.25\). Add this to both side of the equation to complete the square, the equation becomes \((x-4.5)^{2}=4.25\).
03

Solving for x

Taking the square root of both sides, we get \(x-4.5=\pm\sqrt{4.25}\). So, \(x=4.5\pm\sqrt{4.25}\). This gives us two solutions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Equation
A quadratic equation is a polynomial equation of degree 2, usually written in the standard form: \[ ax^2 + bx + c = 0 \]where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). In the quadratic equation \(x^2 - 9x + 16 = 0\), our coefficients are \(a = 1\), \(b = -9\), and \(c = 16\). Quadratic equations can be solved using various methods such as factoring, using the quadratic formula, or completing the square.
  • Factoring: Expressing the equation as a product of two binomials set to zero.
  • Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
  • Completing the Square: Rewriting the equation in the form \((x-h)^2 = k\).
Understanding these methods ensures you can solve quadratics under various conditions. Completing the square, used here, transforms the original quadratic into an equation that is easy to solve using square roots.
Square Roots
Square roots are fundamental in the solution of quadratic equations when using methods like completing the square. A square root of a number \( n \) is a number \( x \) such that \( x^2 = n \). In our equation \((x-4.5)^2 = 4.25\), taking the square root helps to solve for \(x\).
  • The square root symbol is \(\sqrt{}\).
  • Positive and Negative Roots: Mathematical equations can have both \(+\) and \(-\) roots.
  • For instance, from \( (x-4.5)^2 = 4.25 \), taking the square root gives two cases: \( x-4.5 = \pm \sqrt{4.25} \).
This means that \(x\) can take two values: \(4.5 + \sqrt{4.25}\) and \(4.5 - \sqrt{4.25}\), providing the solutions to the equation.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying equations using arithmetic operations and properties of equality. When completing the square with the equation \(x^2 - 9x = -16\), algebraic skills are put to the test. Here's how:
  • Isolating Terms: Move the constant term to the other side to focus on squaring the expression \((x-4.5)^2\).
  • Creating a Perfect Square: Calculate half of \(-9\), then square it to complete the square, which turns the original expression into a perfect square trinomial.
  • Balancing the Equation: Add the squared term \(20.25\) to both sides, maintaining balance in the equation.
By mastering algebraic manipulation, you simplify complex equations into easily solvable forms, especially useful when manual calculations are required.

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Most popular questions from this chapter

Evaluate the function as indicated, and simplify. \(g(x)=2 x^{2}-3 x+1\) (a) \(g(0)\) (b) \(g(-2)\) (c) \(g(1)\) (d) \(g\left(\frac{1}{2}\right)\)

Perform the operation(s). $$(a+b i)(a-b i)$$

Perform the operation and write the result in standard form. $$7 i-(10 i+2)$$

Evaluate the function as indicated, and simplify. \(g(u)=|u+2|\) (a) \(g(2)\) (b) \(g(-2)\) (c) \(g(10)\) (d) \(g\left(-\frac{5}{2}\right)\)

Complex Conjugate What is the complex conjugate of \(i\) ? What is the product of \(i\) and its complex conjugate?
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