Express x2+4x+7 in the form (x+p)2+q. Hence show that equation x2+4x+7=0 has no real root???...I have problem in writing square in my computer so remember x2 is x square and 2 after bracket is squarer too.
the usual way to write powers in this forum is x^2 for x2
so x^2 + 4x + 7
= x^2 + 4x + 4 - 4 + 7
= (x+2)^2 + 3
If I had the parabola y = (x+2)^2 + 3
its vertex would be above the x-axis at (-2,3) and it opens upwards.
So it cannot cross the x-axis.
Since roots are simply the x-intercepts of the corresponding function, we conclude that there are no real roots.
To express the quadratic expression x^2 + 4x + 7 in the form (x + p)^2 + q, we need to complete the square. Here's how you can do it step by step:
Step 1: Divide the coefficient of x by 2 and square the result.
In this case, the coefficient of x is 4. Dividing it by 2 gives us 2, and squaring 2 gives 4.
Step 2: Add the squared result from step 1 to both sides of the equation.
Adding 4 to both sides of the equation x^2 + 4x + 7 = 0, we get:
x^2 + 4x + 7 + 4 = 4
Simplifying, we have:
x^2 + 4x + 11 = 0
Step 3: Rewrite the expression as a perfect square.
To rewrite the expression as a perfect square, we take half of the coefficient of x (which is 2 from step 1) and square it.
(x + 2)^2 = x^2 + 4x + 4
Step 4: Adjust the equation for the difference between the original expression and the perfect square.
We have written (x + 2)^2, but the original expression was x^2 + 4x + 11. The difference between these two expressions is 7, so we need to adjust the equation by subtracting 7 from both sides:
(x + 2)^2 - 7 = 0
Hence, we have expressed x^2 + 4x + 7 in the form (x + p)^2 + q, where p = 2 and q = -7.
To show that the equation x^2 + 4x + 7 = 0 has no real roots, we can use the discriminant:
The discriminant (Δ) is calculated as b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
In this case, a = 1, b = 4, and c = 7. So the discriminant is:
Δ = 4^2 - 4(1)(7)
= 16 - 28
= -12
Since the discriminant is negative (Δ < 0), this means that the equation has no real roots.