Problem 76 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation using any convenient method. $$4 x^{2}=7 x+3$$

Step-by-Step Solution

Answer

The solutions to the quadratic equation are \(x = \frac{7 + \sqrt{97}}{8}\) and \(x = \frac{7 - \sqrt{97}}{8}\)

Step 1: Set the equation to standard form

The first step to solving a quadratic equation via the quadratic formula is to set the equation into standard form. The equation given is \(4x^{2}=7x+3\). Re-arranging the terms so that they equal zero, we get \(4x^{2} - 7x - 3 = 0\)

Step 2: Identify a, b, and c

From the standard form, identify the values for a, b and c. From our equation \(4x^{2}- 7x - 3 = 0\), we can see that \(a = 4\), \(b = -7\), and \(c = -3\)

Step 3: Substitute a, b, and c into the quadratic formula

We now substitute a, b, and c into the formula \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\). Substituting our values into the formula, we get, \(x = \frac{-(-7) \pm \sqrt{(-7)^{2} - 4*4*(-3)}}{2*4}\). Simplifying, we get, \(x = \frac{7 \pm \sqrt{49 - (-48)}}{8} = \frac{7 \pm \sqrt{97}}{8}\)

Step 4: Simplify to find x

By simplifying the equation, we can find the values for x. We have two solutions for x, \(x = \frac{7 + \sqrt{97}}{8}\) and \(x = \frac{7 - \sqrt{97}}{8}\)

Key Concepts

quadratic formula

The quadratic formula is a vital tool in algebra for finding the solutions to quadratic equations. These equations generally take the form of \(ax^2 + bx + c = 0\), where \(a eq 0\). The formula itself is expressed as:

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

This formula helps to find the roots of any quadratic equation. The "\(\pm\)" symbol indicates that there will be two solutions, because we are considering both positive and negative square roots.
Some roots may be complex if the discriminant (\(b^2 - 4ac\)) is negative. The discriminant determines the nature of the roots, such as whether they are real and distinct, real and equal, or complex.

standard form

The standard form of a quadratic equation arranges all terms to one side of the equation, usually expressed as \(ax^2 + bx + c = 0\). In this form:

• \(a\) represents the coefficient of \(x^2\)
• \(b\) is the coefficient of \(x\)
• \(c\) is the constant term

It is crucial to write the equation in this form to correctly identify the coefficients for use in the quadratic formula.

For instance, our equation \(4x^2 - 7x - 3 = 0\) is already in standard form, making it straightforward to define \(a = 4\), \(b = -7\), and \(c = -3\). Once in standard form, you can quickly transition to solving the problem using tools like the quadratic formula.

solving equations

Solving quadratic equations involves finding the values of \(x\) that make the equation true. These values are known as the roots of the equation. The process of solving:

• Start by rearranging the equation into standard form if it is not already.
• Identify \(a\), \(b\), and \(c\) based on the standard form.
• Substitute those into the quadratic formula to solve for \(x\).

Simplifying the results gives the final values of \(x\).
For example, the equation \(4x^2 - 7x - 3 = 0\) gets solved by substituting \(a = 4\), \(b = -7\), and \(c = -3\) into the quadratic formula, leading to the solutions \(x = \frac{7 \pm \sqrt{97}}{8}\). Therefore, we have two possible values for \(x\), demonstrating how each part of the formula works together to produce the solution.

roots of equations

The roots of a quadratic equation are the solutions to the equation \(ax^2 + bx + c = 0\). These roots tell us at what values \(x\) will satisfy the equation.
In our context, the roots can be found using the quadratic formula. Once calculated, the roots can be represented as:

• \(x = \frac{7 + \sqrt{97}}{8}\)
• \(x = \frac{7 - \sqrt{97}}{8}\)

These roots may be real, repeated, or complex depending on the discriminant.
The calculation of the roots provides insight into the nature of the quadratic equation and where it crosses the x-axis on a graph. Understanding these helps to interpret broader patterns and behaviors in algebra.