Question

Find the real solutions, if any, of each equation. Use any method. $$ x^{2}+x=4 $$

Short answer

\( x = \frac{-1 + \sqrt{17}}{2} \) or \( x = \frac{-1 - \sqrt{17}}{2} \)

Step-by-step solution

- Move All Terms to One Side

Start by moving all terms to one side to set the equation to zero. Subtract 4 from both sides:\[ x^2 + x - 4 = 0 \]

- Identify Coefficients for Quadratic Formula

The equation is now in the standard form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = 1 \), and \( c = -4 \).

- Apply the Quadratic Formula

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with the identified coefficients.\[ x = \frac{-(1) \pm \sqrt{(1)^2 - 4(1)(-4)}}{2(1)} \]Simplify inside the square root:\[ x = \frac{-1 \pm \sqrt{1 + 16}}{2} = \frac{-1 \pm \sqrt{17}}{2} \]

- Simplify the Two Solutions

Finally, solve for the two possible values of \( x \):\[ x = \frac{-1 + \sqrt{17}}{2} \]and\[ x = \frac{-1 - \sqrt{17}}{2} \]

Key concepts

Quadratic Formula

The quadratic formula is a powerful tool to solve quadratic equations. It provides a direct way to find the solutions. The quadratic formula is given by: ' x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ' Here: \[ -b \text{ represents the term that changes the sign of } b \] \[ \sqrt{b^2 - 4ac} \text{ represents the discriminant } \] \[ 2a \text{ represents the division by double the leading coefficient} \] The '±' symbol means there will generally be two solutions: one for the addition and one for the subtraction. This way, the quadratic formula captures both possible solutions of the quadratic equation.

Standard Form of a Quadratic Equation

To use the quadratic formula, the quadratic equation must be in its standard form. The standard form of a quadratic equation looks like this: ' ax^2 + bx + c = 0 ' This form has three parts: \[ a \text{ is the coefficient of } x^2 \] \[ b \text{ is the coefficient of } x \] \[ c \text{ is the constant term} \] By setting the equation to zero, we can identify the coefficients necessary to use the quadratic formula. In our exercise, the original equation was 'x^{2} + x = 4'. To set it to zero, we moved the 4 to the left side, resulting in 'x^2 + x - 4 = 0'.

Identifying Coefficients

When you have the equation in standard form, the coefficients can be easily identified. These coefficients are critical for applying the quadratic formula. In the equation ' x^2 + x - 4 = 0 ': \[ a = 1 \text{ (coefficient of } x^2 \text{) } \] \[ b = 1 \text{ (coefficient of } x \text{) } \] \[ c = -4 \text{ (constant term)} \] Once these coefficients are identified, they can be plugged into the quadratic formula to find the solutions. In summary, proper identification and arrangement of these coefficients play a crucial role in solving the quadratic equation accurately!