Question

Find the real solutions, if any, of each equation. $$ 4 y^{2}-8 y=3 $$

Short answer

The solutions are \(y = 1 + \frac{\sqrt{7}}{2}\) and \(y = 1 - \frac{\sqrt{7}}{2}\).

Step-by-step solution

Rewrite the Equation in Standard Form

First, rewrite the given equation in standard quadratic form: \[4y^2 - 8y = 3\]Subtract 3 from both sides to get: \[4y^2 - 8y - 3 = 0\]

Identify Coefficients

Identify the coefficients from the quadratic equation in the form \[ax^2 + bx + c = 0\]. Here, \[a = 4\], \[b = -8\], and \[c = -3\].

Apply the Quadratic Formula

Use the quadratic formula \[y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] to find the values of y. Plug in the coefficients:\[y = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(4)(-3)}}{2(4)}\]Simplify inside the square root first: \[y = \frac{8 \pm \sqrt{64 + 48}}{8}\]\[y = \frac{8 \pm \sqrt{112}}{8}\]

Simplify the Square Root

Simplify \(\sqrt{112}\) as follows: \[\sqrt{112} = \sqrt{16 \cdot 7} = 4\sqrt{7}\]. Plugging this back, we get: \[y = \frac{8 \pm 4\sqrt{7}}{8}\].

Simplify the Solutions

Simplify the fractions: \[y = 1 \pm \frac{\sqrt{7}}{2}\]Therefore, the solutions are: \[y = 1 + \frac{\sqrt{7}}{2}\] and \[y = 1 - \frac{\sqrt{7}}{2}\]

Key concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. Quadratic equations are in the form \[ ax^2 + bx + c = 0\]. The quadratic formula to find the solutions is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

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

The symbol \( \pm \) means we need to calculate two values: one with addition and one with subtraction. These two values are the solutions to the quadratic equation.

Standard Form

The standard form of a quadratic equation is \[ ax^2 + bx + c = 0\]. To solve any quadratic equation using the quadratic formula, you first need to rewrite it in this form. For example, given the equation \[ 4y^2 - 8y = 3\]:1. Subtract 3 from both sides to get:\[ 4y^2 - 8y - 3 = 0\]Now the equation is in standard form where \( a = 4 \), \( b = -8 \), and \( c = -3 \). This step is crucial as it prepares the equation for the application of the quadratic formula.

Simplifying Square Roots

When working with quadratic equations, you will often need to simplify square roots. This process involves finding the prime factors of the number inside the square root and simplifying. For example, simplify \( \sqrt{112} \):

• Factorize 112 as \( 112 = 16 \cdot 7 \)
• Take the square root of each factor:\( \sqrt{16 \cdot 7} = \sqrt{16} \cdot \sqrt{7} \)
• Simplify further as \( \sqrt{16} = 4 \), so the result is \( 4\sqrt{7} \)

This simplified form can then be used in further calculations, like when completing the quadratic formula.

Identifying Coefficients

Before using the quadratic formula, it's essential to correctly identify the coefficients of the quadratic equation in standard form. The standard form is \[ ax^2 + bx + c = 0\], and you need to find the values of \( a \), \( b \), and \( c \).For instance, in the equation \[ 4y^2 - 8y - 3 = 0 \]:

• The coefficient \( a \) (of \( y^2 \)) is 4
• The coefficient \( b \) (of \( y \)) is -8
• The constant term \( c \) is -3

Accurately identifying these values allows you to correctly substitute them into the quadratic formula and solve the equation correctly.