Question

find the roots of quadratic equation x^2+(a-3)x-4a-3

Answer 1

EXPLANATION.

→ x² + ( a - 3 )x - 4a - 3 = 0.

→ x² + ( a - 3 )x - ( 4a + 3) = 0.

→ For equal roots D = 0 or b² - 4ac = 0.

→ ( a - 3)² + 4(1)(4a + 3) = 0.

→ a² + 9 - 6a + 16a + 12 = 0.

→ a² + 10a + 21 = 0

→ a² + 7a + 3a + 21 = 0.

→ a ( a + 7 ) + 3 ( a + 7 ) = 0.

→ ( a + 3 ) ( a + 7 ) = 0.

→ a = -3 and a = -7.

More information.

D = 0 Or [ b² - 4ac = 0 ]

roots are real and equal.

D < 0 or [ b² - 4ac < 0 ].

roots are imaginary.

D > 0 Or [ b² - 4ac > 0 ].

roots are real and unequal.

Answer 2

Question

${ \boxed { \boxed{ \boxed{ \rm \: find \: the \: roots \: of \: quadratic \: equation x^2+(a-3)x-4a-3\: \: }}}}$

Answer

$\mathfrak{ \implies \: {x}^{2} + (a - 3)x - 4a - 3 = 0 } \\ \\ \\ \implies \mathfrak{ {x}^{2} + (a - 3)x - (4a + 3) = 0} \\ \\ \\ \boxed{ \tt \: for \: \: equal \: \: roots \: \: d = 0 } \\ or \\ \boxed{ \tt{b}^{2} - 4ac = 0 } \\ \\ \\ \mathfrak{ \leadsto \: (a - 3)^{2} + 4(1)(4a + 3) = 0} \\ \\ \\ \mathfrak{ \leadsto \: {a}^{2} + 9 - 6a + 16a + 12 = 0 } \\ \\ \\ \mathfrak{ \leadsto \: {a}^{2} + 7a + 3a + 21 = 0 } \\ \\ \\ \mathfrak{ \leadsto \: a(a + 7) + 3(a + 7) = 0} \\ \\ \\ \mathfrak{ \leadsto \: a(a + 3)(a + 7) = 0} \\ \\ \\ \\ { \boxed{ \boxed{ \boxed { \boxed{ \boxed{ \mathfrak{a = - 3 \: and \: \: a = - 7}}}}}}}$

Additional information

In algebra, a quadratic equation (from the Latin quadratus for "square") is any equation that can be rearranged in standard form as

x=-b±√b²-4ac/2a

ax²+bx+c=0

where x represents an unknown, and a, b, and c represent known numbers, where a ≠ 0. If a = 0, then the equation is linear, not quadratic, as there is no ax^2 term. The numbers a, b, and c are the coefficients of the equation and may be distinguished by calling them, respectively, the quadratic coefficient, the linear coefficient and the constant or free term.

The values of x that satisfy the equation are called solutions of the equation, and roots or zeros of the expression on its left-hand side. A quadratic equation has at most two solutions. If there is no real solution, there are two complex solutions. If there is only one solution, one says that it is a double root. A quadratic equation always has two roots, if complex roots are included and a double root is counted for two. A quadratic equation can be factored into an equivalent equation

ax^2+bx+c=a(x-r)(x-s)=0