Question

decide which part of the quadratic formula tells you whether the quadratic equation can be solved by factoring. -b b2 - 4ac 2a use the part of the quadratic formula that you choose above and find its value given the quadratic equation: 2x2 + 7x + 3 = 0

Answer 1

it might be -4ac i'm not sure

Answer 2

recall the quadratic formula is


`\bf ~~~~~~~~~~~~\textit{quadratic formula} \\\\ y=\stackrel{\stackrel{a}{\downarrow }}{a}x^2\stackrel{\stackrel{b}{\downarrow }}{+b}x\stackrel{\stackrel{c}{\downarrow }}{+c} \qquad \qquad x= \cfrac{ - b \pm \sqrt { b^2 -4 a c}}{2 a}`

and the tell-tale part is the discriminant, namely the radicand above,


`\bf \qquad \qquad \qquad \textit{discriminant of a quadratic} \\\\\\ y=\stackrel{\stackrel{a}{\downarrow }}{a}x^2\stackrel{\stackrel{b}{\downarrow }}{+b}x\stackrel{\stackrel{c}{\downarrow }}{+c} ~~~~~~~~ \stackrel{discriminant}{b^2-4ac}= \begin{cases} 0&\textit{one solution}\\ positive&\textit{two solutions}\\ negative&\textit{no solution} \end{cases}`

so, we can check its discriminant, and if it's negative, then we know it has no solutions, if is positive, then we know we can factor it.

lets check it anyway.


`\bf \qquad \qquad \qquad \textit{discriminant of a quadratic} \\\\\\ \stackrel{\stackrel{a}{\downarrow }}{2}x^2\stackrel{\stackrel{b}{\downarrow }}{+7}x\stackrel{\stackrel{c}{\downarrow }}{+3}=0 ~~~~~~~~ \stackrel{discriminant}{b^2-4ac}= \begin{cases} 0&\textit{one solution}\\ positive&\textit{two solutions}\\ negative&\textit{no solution} \end{cases} \\\\\\ (7)^2-4(2)(3)\implies 49-24\implies 25\impliedby \textit{is positive}`


`\bf ~~~~~~~~~~~~\textit{quadratic formula} \\\\ \stackrel{\stackrel{a}{\downarrow }}{2}x^2\stackrel{\stackrel{b}{\downarrow }}{+7}x\stackrel{\stackrel{c}{\downarrow }}{+3}=0 \qquad \qquad x= \cfrac{ - b \pm \sqrt { b^2 -4 a c}}{2 a} \\\\\\ x=\cfrac{-7\pm√(25)}{2(2)}\implies x=\cfrac{-7\pm 5}{4}\implies x= \begin{cases} -\cfrac{2}{4}\implies &-\cfrac{1}{2}\\\\ \cfrac{-12}{4}\implies &-3 \end{cases}`