Problem 71 - Algebra and Trigonometry

Question

Solve equation using the quadratic formula. $$ 4 x^{2}=2 x+7 $$

Step-by-Step Solution

Answer

The solutions for the quadratic equation are \(x_{1} = 1.5\) and \(x_{2} = -1.17\).

Step 1: Rewrite the equation in the standard form

First, the equation must be written in the standard form of a quadratic equation. To do this, subtract \(2x + 7\) from both sides of the given equation. The equation then becomes \(4x^{2} - 2x - 7 = 0\). Now, from the rewritten equation we can see that \(a = 4\), \(b = -2\), and \(c = -7\).

Step 2: Calculate the discriminant

The discriminant is part of the quadratic formula and is represented as \(b^{2} - 4ac\). The discriminant tells us the number and type of solutions our quadratic equation has. Calculate the discriminant using \(a = 4\), \(b = -2\), and \(c = -7\). This results in a discriminant of \((-2)^{2} - 4*4*(-7) = 116\).

Step 3: Solve the quadratic equation with the quadratic formula

Finally, substitute \(a = 4\), \(b = -2\), and the calculated discriminant into the quadratic formula \((-b \pm \sqrt{b^{2} - 4ac}) /2a\). This results into two real solutions \(x_{1} = \frac{2 + \sqrt{116}}{8} = 1.5\) and \(x_{2} = \frac{2 - \sqrt{116}}{8} = -1.17\).

Key Concepts

Discriminant

In quadratic equations, the discriminant is a crucial component found in the quadratic formula. It's represented as \( b^2 - 4ac \), where \( a \), \( b \), and \( c \) are coefficients from the standard form of a quadratic equation \( ax^2 + bx + c = 0 \). The role of the discriminant is to determine the nature and quantity of solutions an equation has.

• If the discriminant is positive, the quadratic equation has two distinct real solutions.
• If it is zero, there is exactly one real solution, which means the parabola touches the x-axis at a single point.
• If the discriminant is negative, the solutions are complex or imaginary, indicating that the parabola does not intersect the x-axis.

Understanding the value of the discriminant helps predict how the graph of a quadratic function behaves and what kind of solutions we can expect.

Standard Form of a Quadratic Equation

To effectively solve a quadratic equation using the quadratic formula, it's crucial to express the equation in its standard form, which is \( ax^2 + bx + c = 0 \). This form organizes all terms of the equation, making it easier to identify the coefficients needed for solving.
In the given exercise, the original equation was \( 4x^2 = 2x + 7 \). To put this equation in the standard form:

• Subtract \(2x + 7\) from both sides to obtain \(4x^2 - 2x - 7 = 0\).
• Now, the coefficients \(a\), \(b\), and \(c\) are found: \(a = 4\), \(b = -2\), and \(c = -7\).

Writing a quadratic equation in standard form is a simple step but fundamental for accurately applying methods of solutions like the quadratic formula.

Solving Quadratic Equations

Solving quadratic equations often involves using the quadratic formula, especially when the equation cannot be easily factored. The quadratic formula is defined as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
Steps to Solve:

• First, ensure the equation is in standard form \( ax^2 + bx + c = 0 \).
• Identify and substitute the coefficients \(a\), \(b\), and \(c\) into the formula.
• Calculate the discriminant, \(b^2 - 4ac\), to determine the nature of the roots.
• Perform the calculation within the square root and continue simplifying.
• Evaluate both \(+\) and \(-\) versions of the formula to find two possible solutions, if applicable.

For this exercise, with the standard form \(4x^2 - 2x - 7 = 0\), substituting \(a = 4\), \(b = -2\), and \(c = -7\) yields two real solutions: \(x_1 = 1.5\) and \(x_2 \approx -1.17\).
This formula provides a systematic way to find solutions for any quadratic equation.