Problem 73 - Elementary Algebra

Question

Problem: Use the quadratic formula to solve \(2 x^{2}-51 x+270=0\) Incorrect Answer: \(x=\frac{-51 \pm \sqrt{(-51)^{2}-4(2)(270)}}{2(2)}\) \(x=\frac{-51 \pm \sqrt{2601-2160}}{4}\) \(x=\frac{-51 \pm \sqrt{441}}{4}\) \(x=\frac{-51 \pm 21}{4}\) \begin{array}{lll} x=\frac{-51+21}{4} & \text { or } & x=\frac{-51-21}{4} \\ x=\frac{-30}{4} & \text { or } & x=\frac{-72}{4} \\ x=-\frac{15}{2} & \text { or } & x=-18 \end{array}

Step-by-Step Solution

Answer

The solutions are \(x = 18\) and \(x = 7.5\).

Step 1: Identify the quadratic formula

Recall the quadratic formula which is given by: \[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\]Here, the equation is in the form of \(ax^2 + bx + c = 0\).

Step 2: Determine coefficients

Identify coefficients from the given equation \(2x^2 - 51x + 270 = 0\).Here, \(a = 2\), \(b = -51\), and \(c = 270\).

Step 3: Calculate the discriminant

The discriminant is calculated as \(b^2 - 4ac\):\[(-51)^2 - 4(2)(270)\]Calculate each part:\[2601 - 2160 = 441\]

Step 4: Apply the quadratic formula

Substitute the coefficients and the discriminant into the quadratic formula:\[x = \frac{-(-51) \pm \sqrt{441}}{2(2)}\]Simplifying inside of the square root and the fraction:\[x = \frac{51 \pm 21}{4}\]

Step 5: Calculate the solutions

Solve for the two possible values of \(x\):First solution:\[x = \frac{51 + 21}{4} = \frac{72}{4} = 18\]Second solution:\[x = \frac{51 - 21}{4} = \frac{30}{4} = 7.5\]

Key Concepts

quadratic equations

Quadratic equations are polynomial equations of degree 2. They take the form: \[ax^2 + bx + c = 0\]where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The highest exponent of the variable \(x\) is 2, making it a quadratic equation.

A quadratic equation might look like \(2x^2 - 51x + 270 = 0\). Here,

• a is 2
• b is -51
• c is 270

To solve a quadratic equation, there are several methods, with the quadratic formula being one of the most reliable approaches because it works for all quadratic equations, regardless of their form.

discriminant

The discriminant is a key part of the quadratic formula and helps determine the nature of the roots of a quadratic equation. It is found inside the square root in the formula: \[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\] The term \(b^2 - 4ac\) is known as the discriminant.

The value of the discriminant can reveal three scenarios:

• If the discriminant is positive (\(b^2 - 4ac > 0\)), the quadratic equation has two distinct real roots.
• If the discriminant is zero (\(b^2 - 4ac = 0\)), the equation has exactly one real root, known as a repeated or double root.
• If the discriminant is negative (\(b^2 - 4ac < 0\)), the equation has no real roots but two complex conjugate roots.

In the example equation \(2x^2 - 51x + 270 = 0\), the discriminant is calculated as: \[(-51)^2 - 4(2)(270) = 2601 - 2160 = 441\]Since 441 is positive, it indicates that the equation has two distinct real roots.

solving equations

Solving quadratic equations often involves finding the values of \(x\) that satisfy the equation. For many equations, the quadratic formula is an effective method: \[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\]Here’s a step-by-step guide to solving the example equation \(2x^2 - 51x + 270 = 0\):1. **Identify the coefficients:** Given the equation, \(a = 2\), \(b = -51\), and \(c = 270\).
2. **Calculate the discriminant:** The discriminant is \(b^2 - 4ac\). Calculate: \[(-51)^2 - 4(2)(270) = 441\]3. **Apply the quadratic formula:** Substitute \(a\), \(b\), and the discriminant into the quadratic formula: \[x = \frac{-(-51) \pm \sqrt{441}}{2(2)} = \frac{51 \pm 21}{4}\]Simplify the equation to find the potential values for \(x\).
4. **Find the solutions:** Calculate the possible values of \(x\): \[x = \frac{51 + 21}{4} = 18\]\[x = \frac{51 - 21}{4} = 7.5\]So, the solutions to the quadratic equation \(2x^2 - 51x + 270 = 0\) are \(x = 18\) and \(x = 7.5\).

By following these steps carefully, you can solve almost any quadratic equation efficiently.