# Question Video: Solving Quadratic Inequalities in One Variable Mathematics • 10th Grade

Write the interval describing all solutions to the inequality −𝑥² − 2𝑥 + 168 ≥ 0.

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### Video Transcript

Write the interval describing all solutions to the inequality negative 𝑥 squared minus two 𝑥 plus 168 is greater than or equal to zero.

In order to solve any quadratic inequality, we begin by solving the equivalent equation, in this case, negative 𝑥 squared minus two 𝑥 plus 168 is equal to zero. As our coefficient of 𝑥 squared is negative, it will be easier to factor this equation by multiplying it all by negative one. This gives us 𝑥 squared plus two 𝑥 minus 168. The signs of all three terms are now the opposite. We can factor this equation into two sets of parentheses or brackets by looking for two numbers with a product of negative 168 and a sum of positive two.

14 multiplied by negative 12 is negative 168, and 14 plus negative 12 is equal to two. Our two sets of parentheses are 𝑥 plus 14 and 𝑥 minus 12. When 𝑥 plus 14 is equal to zero, 𝑥 is equal to negative 14. Likewise, when 𝑥 minus 12 is equal to zero, 𝑥 is equal to 12. The equation negative 𝑥 squared minus two 𝑥 plus 168 equals zero has two solutions, 𝑥 equals negative 14 and 𝑥 equals 12. We know that when the coefficient of 𝑥 squared is positive, any quadratic function will be a u-shaped parabola. When the coefficient is negative, as in this case, we will have an n-shaped parabola.

The equation 𝑦 equals negative 𝑥 squared minus two 𝑥 plus 168 will look as shown in the graph. It has a 𝑦-intercept at 168 and intersects the 𝑥-axis at negative 14 and 12. We want the solutions where this graph is greater than or equal to zero. This is all the points that are above or on the 𝑥-axis. These occur between the 𝑥-values of negative 14 and 12. The solution can therefore be written as the set of values between negative 14 and 12 inclusive. We denote this using square brackets.