Problem 30 - Elementary Algebra

Question

\(10 h^{2}+11 h-13=0\)

Step-by-Step Solution

Answer

The roots are \(h = \frac{-11 + \sqrt{641}}{20}\) and \(h = \frac{-11 - \sqrt{641}}{20}\).

Step 1 - Identify coefficients

In the quadratic equation, identify the coefficients for the terms. The given equation is: \(10h^{2} + 11h - 13 = 0\)Here, the coefficient for \(h^{2}\) is \(a = 10\), for \(h\) is \(b = 11\), and the constant term is \(c = -13\).

Step 2 - Use the quadratic formula

The quadratic formula to find the roots of a quadratic equation \(ax^{2} + bx + c = 0\) is given by: \[h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Substitute the values of \(a\), \(b\), and \(c\) into the formula.

Step 3 - Calculate discriminant

Calculate the discriminant \(\Delta\) using the formula: \(\Delta = b^{2} - 4ac\)Substitute the values: \(\Delta = 11^{2} - 4 \cdot 10 \cdot (-13) = 121 + 520 = 641\).

Step 4 - Find the square root of the discriminant

Compute the square root of the discriminant: \(\sqrt{641}\).

Step 5 - Compute the roots

Substitute the values and compute the roots using the quadratic formula: \[h = \frac{-11 \pm \sqrt{641}}{20}\]This gives the solutions: \[h = \frac{-11 + \sqrt{641}}{20}\] and \[h = \frac{-11 - \sqrt{641}}{20}\].

Key Concepts

quadratic formula

The quadratic formula is a crucial tool for solving quadratic equations. A quadratic equation has the general form: ax^2 + bx + c = 0. The quadratic formula provides a way to find the roots (solutions for x) of any quadratic equation. Here's the formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. To use this formula:

• Identify the coefficients a, b, and c from the equation.
• Plug them into the formula.
• Solve for x by performing the arithmetic operations as per the formula.

This powerful formula allows us to find the precise values of x, which satisfy the quadratic equation.

discriminant

The discriminant (\(\Delta\)) is an essential part of the quadratic formula. It provides important information about the nature of the roots. The discriminant is found using the formula: \( \Delta = b^2 - 4ac \). Here's what the value of \(\Delta\) tells us:

• If \(\Delta > 0 \) , there are two distinct real roots.
• If \(\Delta = 0 \) , there is exactly one real root, also known as a repeated or double root.
• If \(\Delta < 0 \) , there are no real roots, but two complex conjugate roots.

In our example, \(\Delta = 641\), which is greater than 0. This means the quadratic equation \(10 h^2 + 11 h - 13 = 0\) has two distinct real roots.

roots of quadratic equation

The roots of a quadratic equation are the values of x that satisfy the equation. Using the quadratic formula, we can find these roots by solving the expression: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. For our specific equation \(10 h^2 + 11h - 13 = 0\), we already calculated the discriminant and substituted all the values into the formula: \[ h = \frac{-11 \pm \sqrt{641}}{20} \]. This gives us the two roots:

• \( h_1 = \frac{-11 + \sqrt{641}}{20} \).
• \( h_2 = \frac{-11 - \sqrt{641}}{20} \).

These roots represent the points where the quadratic equation intersects the x-axis. Once you have the roots, you can conclude the solution for the quadratic equation.