Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter P - P.1 - Graphs and Models - Exercises - Page 9: 80

Answer

This statement is true.

Work Step by Step

To find the intercepts with $x$ axis (if they exist) put $y=0$ in the equation and calculate $x$. The obtained values for $x$ will determine where the graph intercepts the $x$ axis: $$ax^2+bx+c=0.$$ This is a quadratic equation in $x$ and its' solutions are given by quadratic equation formula $$x_{1,2} = \frac{-b\pm\sqrt{b^2-4ac}}{2a}.$$ Since $b^2-4ac=0$ then also $\sqrt{b^2-4ac} = 0$ so in the formula for $x_{1,2}$ we actually have $$x_{1,2} = \frac{-b\pm0}{2a} = -\frac{b}{2a}$$ so there is only one solution since it doesn't matter whether we add or subtract $0$. So the only intercept with the $x$ axis will occur in $$x=-\frac{b}{2a}$$ and the statement is true.
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