Try the fastest way to create flashcards
Question

Identify whether the statement given below is true or false.$\forall$ real numbers $a, b, c, \exists$ exactly two real numbers $x$ such that $a x^2+b x+c=0$. If false, find a counterexample.

Solution

Verified
Step 1
1 of 2

$\text{This statement is a \textcolor{#4257b2}{false}}$

$\text{This statement is not true for all real numbers $a$, $b$ and $c$}$.

$\text{For example, let $a=0$}$

Then, given equation has only one solution:

\begin{align*}\\ 0\cdot x^{2}+b\cdot x+c=0\\ b\cdot x+c=0\\ b \cdot x=-c \tag{\text{subtract c from both sides}}\\ x=-\dfrac{c}{b} \tag{\text{divide both sides by b}}\\ \end{align*}

$\text{So, if $a=0$, then, given equation}$ $a\cdot x^{2}+b\cdot x+c=0$ $\text{has only one solution $x=-\dfrac{c}{b}$}$

Recommended textbook solutions

Discrete Mathematics and Its Applications

7th EditionISBN: 9780073383095 (1 more)Kenneth Rosen
4,283 solutions

Precalculus and Discrete Mathematics

3rd EditionISBN: 9780076214211Anthony L. Peressini, Mary Helen Wiltjer, Molly A. Rockstroh, Peter D. DeCraene, Steven S. Viktora, Ward E. Canfield, Zalman Usiskin
3,203 solutions

Discrete Mathematics and Its Applications

8th EditionISBN: 9781259676512Kenneth Rosen
4,397 solutions

Discrete Mathematics with Applications

5th EditionISBN: 9781337694193 (4 more)Susanna S. Epp
2,641 solutions