Solve Using the Quadratic Formula x^2+x=7x+16
The question provided is a mathematical problem that requires the use of the Quadratic Formula to solve for the variable 'x'. It presents a quadratic equation in the standard form ax^2 + bx + c = 0, with specific coefficients substituted into it, resulting in the equation x^2 + x - 7x - 16 = 0. The task is to find the values of 'x' that satisfy the equation by applying the Quadratic Formula, which is a general method for finding the solutions of quadratic equations.
$x^{2} + x = 7 x + 16$
Answer
Solution:
Step 1
Rearrange the equation to set it to zero.
Step 1.1
Transfer all terms to one side of the equation.
Step 1.1.1
Subtract $7x$ from each side to get $x^2 + x - 7x = 16$.
Step 1.1.2
Also, subtract $16$ to obtain $x^2 - 6x - 16 = 0$.
Step 1.2
Combine like terms to simplify: $x^2 - 6x - 16 = 0$.
Step 2
Apply the quadratic formula: $\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Step 3
Insert $a = 1$, $b = -6$, and $c = -16$ into the formula and compute $x$.
Step 4
Carry out the simplification process.
Step 4.1
Simplify the expression in the numerator.
Step 4.1.1
Square $-6$ to find $36$.
Step 4.1.2
Calculate $4 \cdot 1 \cdot -16$.
Step 4.1.2.1
Multiply $-4$ by $1$.
Step 4.1.2.2
Then multiply $-4$ by $-16$ to get $64$.
Step 4.1.3
Add $36$ to $64$ to get $100$.
Step 4.1.4
Express $100$ as $(10)^2$.
Step 4.1.5
Extract the square root to get $10$.
Step 4.2
Simplify the denominator by multiplying $2$ by $1$.
Step 4.3
Divide $6 \pm 10$ by $2$ to simplify further.
Step 5
State both solutions from the simplified form: $x = 8$ and $x = -2$.
Knowledge Notes:
The quadratic formula is a fundamental tool in algebra for solving quadratic equations, which are of the form $ax^2 + bx + c = 0$. The formula is derived from completing the square and provides the roots of the quadratic equation, where $a$, $b$, and $c$ are coefficients with $a \neq 0$.
The quadratic formula is given by:
$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
The term under the square root, $b^2 - 4ac$, is known as the discriminant. It determines the nature of the roots:
If the discriminant is positive, there are two distinct real roots.
If the discriminant is zero, there is one real root (a repeated root).
If the discriminant is negative, there are two complex roots.
In this problem, we first rearranged the given equation into standard form and then applied the quadratic formula. The discriminant was positive, indicating two real solutions. After simplifying, we found the two roots of the equation.
