Problem 9 - Basic Engineering Mathematics

Question

Solve (a) \(x^{2}+2 x-8=0\) and (b) \(3 x^{2}-11 x-4=0\) by using the quadratic formula.

Step-by-Step Solution

Answer

(a) Solutions: \(x = 2\) and \(x = -4\); (b) Solutions: \(x = 4\) and \(x = -\frac{1}{3}\).

Step 1: Identify Coefficients for Equation (a)

For the quadratic equation \(x^2 + 2x - 8 = 0\), identify the coefficients \(a\), \(b\), and \(c\). Here, \(a = 1\), \(b = 2\), and \(c = -8\).

Step 2: Apply the Quadratic Formula for Equation (a)

The quadratic formula is given by: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute \(a = 1\), \(b = 2\), \(c = -8\) into the formula. \[x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-8)}}{2(1)}\].

Step 3: Simplify Inside the Square Root for Equation (a)

Calculate inside the square root for equation (a): \(2^2 - 4 \times 1 \times (-8) = 4 + 32 = 36\). Thus, the expression becomes \(x = \frac{-2 \pm \sqrt{36}}{2}\).

Step 4: Calculate the Solutions for Equation (a)

Evaluate \(\sqrt{36}\) which is 6. Thus, \(x = \frac{-2 \pm 6}{2}\). Calculate the two possible solutions: \(x_1 = \frac{-2 + 6}{2} = 2\) and \(x_2 = \frac{-2 - 6}{2} = -4\). The solutions are \(x = 2\) and \(x = -4\).

Step 5: Identify Coefficients for Equation (b)

For the quadratic equation \(3x^2 - 11x - 4 = 0\), identify the coefficients \(a\), \(b\), and \(c\). Here, \(a = 3\), \(b = -11\), and \(c = -4\).

Step 6: Apply the Quadratic Formula for Equation (b)

Substitute \(a = 3\), \(b = -11\), \(c = -4\) into the quadratic formula:\[x = \frac{11 \pm \sqrt{(-11)^2 - 4 \cdot 3 \cdot (-4)}}{2 \cdot 3}\].

Step 7: Simplify Inside the Square Root for Equation (b)

Calculate inside the square root: \((-11)^2 - 4 \times 3 \times (-4) = 121 + 48 = 169\). Thus, the expression becomes \(x = \frac{11 \pm \sqrt{169}}{6}\).

Step 8: Calculate the Solutions for Equation (b)

Evaluate \(\sqrt{169}\) which is 13. Thus, \(x = \frac{11 \pm 13}{6}\). Calculate the two possible solutions: \(x_1 = \frac{11 + 13}{6} = 4\) and \(x_2 = \frac{11 - 13}{6} = -\frac{1}{3}\). The solutions are \(x = 4\) and \(x = -\frac{1}{3}\).

Key Concepts

Quadratic Formula

The quadratic formula is a mathematical tool we use to find solutions to quadratic equations. A quadratic equation is a polynomial of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants.
The quadratic formula is expressed as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

It allows us to solve any quadratic equation by plugging in the values of \( a \), \( b \), and \( c \) into the formula.
The term under the square root, \( b^2 - 4ac \), is called the discriminant. The discriminant helps determine the nature of the roots.

• If \( b^2 - 4ac > 0 \): The equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \): There is one real root, or a repeated root.
• If \( b^2 - 4ac < 0 \): The equation has two complex roots.

Using the quadratic formula ensures that you can find solutions even if the equation cannot be easily factored.

Factoring Quadratic Equations

Factoring quadratic equations is another way to find the roots of the equations, but it is not always applicable. Not all quadratic equations can be factored easily with integer solutions.
When factoring is possible, it simplifies the process since you look for two numbers that multiply to \( ac \) (the product of \( a \) and \( c \)) and add to \( b \). The equation \( ax^2 + bx + c \) then rewrites into a product of two binomials:

• \( (px + q)(rx + s) = 0 \)

Each binomial set equal to zero gives the solutions:

• \( px + q = 0 \)
• \( rx + s = 0 \)

Though factoring is quick, it requires the numbers to play nice. If you see that factoring does not fit, then the quadratic formula is your next best friend.
This approach works best when dealing with smaller numbers or cases where common factors are obvious, and it saves calculation time compared to using the quadratic formula.

Solutions of Quadratic Equations

The solutions of a quadratic equation are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). These solutions are often referred to as "roots" of the equation and they can be found using several methods.

One method to find these solutions is by using the quadratic formula discussed earlier. Another method is factoring when applicable, and there's also completing the square, though it is less common for straightforward problems.
Quadratic equations can have:

• Two distinct real solutions.
• One real solution (a repeated or double root).
• Two complex solutions when no real number solutions exist.

Understanding these different scenarios is crucial in graphing these equations, as the roots affect where the parabola intersects the x-axis.Remember, each root represents a point where the equation equals zero, meaning the curve touches or crosses the x-axis here.