Question

consider the quadratic equation ( ax^2 + bx + c = 0 ).
solve the equation ( 2x^2 + 6x + 10 = 0 ) using the quadratic formula ( x = \frac{-b pm sqrt{b^2 - 4ac}}{2a} ).
which statement correctly explains the process and solution for the given quadratic equation?

the coefficients are ( a = 2 ), ( b = 10 ), and ( c = 6 ). the discriminant, ( b^2 - 4ac = 10^2 - 4(2)(6) = 28 ), suggests real and unequal roots. thus the solutions are ( x = -2 ) and ( x = -3 ).

the coefficients are ( a = 6 ), ( b = 2 ), and ( c = 10 ). the discriminant, ( b^2 - 4ac = 2^2 - 4(6)(10) = -236 ), leads to complex solutions. the roots are calculated as ( x = -0.167 pm sqrt{59}i ).

the coefficients are ( a = 2 ), ( b = 6 ), and ( c = 10 ). the discriminant, ( b^2 - 4ac = 6^2 - 4(2)(10) = -44 ), indicates complex roots. thus the formula provides ( x = \frac{-6 pm 2sqrt{11}i}{4} ), using the fact that the square root of ( -44 ) simplifies to ( 2sqrt{11}i ).

the coefficients are ( a = 2 ), ( b = 6 ), and ( c = 10 ). the discriminant,

Explanation:

Step1: Identify coefficients

For $2x^2 + 6x + 10 = 0$, $a=2$, $b=6$, $c=10$

Step2: Calculate discriminant

$\Delta = b^2 - 4ac = 6^2 - 4(2)(10) = 36 - 80 = -44$

Step3: Simplify square root of discriminant

$\sqrt{-44} = \sqrt{4 \times (-11)} = 2\sqrt{11}i$

Step4: Apply quadratic formula

$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} = \frac{-6 \pm 2\sqrt{11}i}{4}$

Answer:

The coefficients are $a = 2$, $b = 6$, and $c = 10$. The discriminant, $b^2 - 4ac = 6^2 - 4(2)(10) = -44$, indicates complex roots. Thus the formula provides $x = \frac{-6\pm2\sqrt{11}i}{4}$, using the fact that the square root of $-44$ simplifies to $2\sqrt{11}i$.