Question Video: Solving Quadratic Inequalities in One Variable | Nagwa Question Video: Solving Quadratic Inequalities in One Variable | Nagwa

Question Video: Solving Quadratic Inequalities in One Variable Mathematics • First Year of Secondary School

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Find the interval containing all solutions to the inequality 64 − 𝑥² < 0.

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Video Transcript

Find the interval containing all solutions to the inequality 64 minus 𝑥 squared is less than zero.

In order to solve any quadratic inequality, we begin by solving the equivalent quadratic equation. In this case, 64 minus 𝑥 squared is equal to zero. We might notice at this stage that this equation is the difference of two squares. It therefore factors into two sets of parentheses, eight plus 𝑥 and eight minus 𝑥. The products of these two sets of parentheses will be equal to zero when one of the sets equals zero. Either eight plus 𝑥 equals zero or eight minus 𝑥 equals zero. This gives us two solutions: 𝑥 equals negative eight or 𝑥 equals eight.

An alternative method to solve the quadratic equation would be to add 𝑥 squared to both sides. We could then square root both sides of this equation, giving us 𝑥 equals positive or negative eight, as the square root of 64 is eight. These two solutions tell us that the graph of the quadratic equation 𝑦 equals 64 minus 𝑥 squared cross the 𝑥-axis at two points: 𝑥 equals negative eight and 𝑥 equals eight. Any quadratic equation will be a u-shaped parabola when the coefficient of 𝑥 squared is positive. In this question, we have a negative coefficient of 𝑥 squared. Therefore, our parabola will be n-shaped.

The equation 𝑦 equals 64 minus 𝑥 squared will have a 𝑦-intercept at 64. As already mentioned, it will cross the 𝑥-axis at negative eight and eight. The graph of the function will be as shown. We want this equation to be less than zero. This is all the points below the 𝑥-axis. This will be true for all 𝑥-values that are less than negative eight or greater than positive eight.

One way of writing our solution would be the set of values from negative ∞ to eight or the set of values from positive ∞ to eight. This could also be written as the set of all real values minus those between negative eight and eight inclusive. Either of these would be acceptable answers for this question.

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