Solve by using the Quadratic Formula. 16 c^2+24 c+9=0 |...

Name: Problem 117, Chapter 10: Elementary Algebra
Uploaded: 2021-07-20T09:35:37-08:00
Duration: 3 min 54 s
Channel: Sreeraj P
Description: Solve by using the Quadratic Formula. 16 c^2+24 c+9=0

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants and a is nonzero. It forms the basis for many algebraic techniques and solutions in algebra.

Standard Form

Writing a quadratic equation in the standard form ax² + bx + c = 0 is essential because it clearly identifies the coefficients needed for applying methods such as the quadratic formula or factoring.

Quadratic Formula

The quadratic formula, x = (-b ± ?(b² - 4ac)) / (2a), is a general solution method for quadratic equations. It allows for the determination of the roots by directly substituting the coefficients from the standard form.

Discriminant

The discriminant, given by b² - 4ac, is a crucial component in the quadratic formula that helps determine the nature of the roots. Depending on whether the discriminant is positive, zero, or negative, the quadratic equation will have two distinct real roots, one repeated real root, or two complex conjugate roots, respectively.

Transcript

00:01 We have the quadratic equation 16 c squared plus 24 c plus 9 equal 0 and we have to find a solution by using the quadratic formula so for quadratic formula if you have a general quadratic equation of the form a x square plus bx plus c is equal to well a is not zero a cannot be zero the quadratic formula gives the solution by x is equal to minus b plus or minus root of b square minus 4a c all divided by 2a where values of a b and c are taken from the quadratic equation so here the format of the given question and the general equation is the same so we just have to extract this value that is abc the co -options and just substitute in this formula so for that we have a b and c and the values of ab and c from the values of ab and c from the question is a is 16 b is 24 and c is 9 so substituting those into this we'll get x is equal to minus b that is minus 24 plus or minus root of b square that is 24 square plus so sorry 24 square minus 24 square minus 4 multiplied by a is 16 multiplied by c is 9 whole divided by 2a plus 2 multiplied by 16 right so this will become minus 24 plus or minus root of 24 square so you have to find 21 square so 24 square is 576 minus 4 multiplied by 16 multiplied by 9 4 multiplied by 16 multiplied by 9 will give you 576 so, 176, well divided by 2 multiple by 16, that is 32.

02:18 So here, here we have 576 minus 576, right? so what does that mean that b square minus e? sorry, b square minus 4 ac is 0 or the discriminant is 0.

02:33 So this turns out to be 0.

02:37 So in case where discriminant is 0, we just have one solution for the quadratory.

02:44 Equation right ideally we have two solutions but both the solutions are the same so that means this becomes x is equal to minus 24 plus or minus 0 will be the right but it makes no difference so it becomes minus 24 by 32 now to cut the common factors from cancel out the common factors from the fraction we'll get to my minus 24 can be written by minus 3 multiplied by 8 right and 32 can be 4 multiplied by 8.

03:17 So when you cancel these to 8, we'll get minus 3 by 4.

03:22 So that means x is equal to minus 3 by 4...

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