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Sometimes, quadratic equations have no solutions that can be expressed using real numbers. However, it is possible to use complex numbers to express these non-real solutions.

The quadratic formula is $x=2a-b±b_{2}−4ac ,$

where $a,b,$ and $c$ correspond with the values of a quadratic equation written in standard form, $ax_{2}+bx+c=0.$ It is derived by completing the square on the general standard form equation. The quadratic formula can be used to find solution(s) to quadratic equations.The solutions of a quadratic equation in the form $ax_{2}+bx+c=0$ can be interpreted graphically as the zeros of the quadratic function $y=ax_{2}+bx+c.$

If the function hasIn the quadratic formula, the term under the radical sign is called the discriminant.

It's possible to use the discriminant to determine the number of solutions a quadratic equation has. $b_{2}−4acb_{2}−4acb_{2}−4ac >0⇔2real solutions=0⇔1real solution<0⇔0real solutions $ The solutions to a quadratic equation correspond to the zeros of the parabola.

When the discriminant is negative, there are no real solutions of the quadratic equation. However, the square root of a negative number can be written as an imaginary number. This way, complex solutions of a quadratic equation can be found.

Solve the equation using the quadratic formula. $2x_{2}−4x+10=0$

Show Solution

Since the quadratic equation is given in standard form, we can immediately identify the constants
$a=2,b=-4,andc=10.$
Let's substitute these values into the quadratic formula to solve the equation.
Notice that there is a negative number inside the square root. Thus, the equation has no real solutions. However, we can calculate the complex roots using the identity
$-a =a ⋅i.$
The solutions to the equation are $x=1+2i$ and $x=1−2i.$

$2x_{2}−4x+10=0$

UseQuadFormUse the Quadratic Formula: $a=2,b=-4,c=10$

$x=2⋅2-(-4)±(-4)_{2}−4⋅2⋅10 $

NegNeg$-(-a)=a$

$x=2⋅24±(-4)_{2}−4⋅2⋅10 $

CalcPowProdCalculate power and product

$x=44±16−80 $

SubTermSubtract term

$x=44±-64 $

$x=44±-64 $

SqrtNegToSqrtI$-a =a ⋅i$

$x=44±64 ⋅i $

CalcRootCalculate root

$x=44±8i $

SimpQuotSimplify quotient

$x=1±2i$

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