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Find the roots of the quadratic equation \(x^23x=0 \).
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Are \(x=0,\ x=1 \) the solution of the equation \(x^2+x+1=0 \)?
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If
If
So are not the solution
If \(4a^2x^24abx+k=0 \) has equal roots, then find the value of \(k \).
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Find the discriminant of the quadratic equation \(bx^22\sqrt{ac}x+b=0 \)
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If 2 is a root of the equation \(x^2+bx+12=0 \) and the equation has equal roots, find the value of b,
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If one of the roots of the equation \(3a^2x^2+8abx+4b^2=0 \) is \(\dfrac{2b}{a} \), then find the other root.
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Let other root be
If \(x=\dfrac{c}{b} \) is a root of \(abx^2+(b^2ac)\ xk=0 \), then find the value of \( k\).
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State whether the quad ratic equation \((x\sqrt2)^2\sqrt2(x+1)=0 \) has two distinct real roots. Justify your answer.
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so the roots are real and distinct
If \(b=0,\ c<0 \), is it true that the roots of the quadratic equation \(x^2+bx+c=0 \) are numerically equal and opposite in sign? Justify.
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Yes the roots are equal and opposite of the two roots are and
as then . This is only possible when and and equal and opposite
Does \((x1)^2+2(x+1)=0 \) have a real root? Justify your answer.
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This is not possible as under root of a negetive number leads to imaginary number has no real roots
Solve the following quadratic equations by factorisation:
\(x^2(\sqrt2+1)x+\sqrt2=0 \)
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Solve the following quadratic equations by factorisation:
\(4x^2 + 4bx – (a^2 – b^2) = 0 \)
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Solve the following quadratic equations by factorisation:
\(ax^2 + (4a^2 – 3b) x – 12ab = 0 \)
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Solve the following quadratic equations by factorisation:
\(\dfrac{m}{n}x^2+\dfrac{n}{m}=12x \)
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Solve the following quadratic equations by factorisation:
\((a+b)^2x^24abx(ab)^2=0 \)
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Solve the following quadratic equations by factorisation:
\(x^2 – x – a (a + 1) = 0 \)
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Solve the following quadratic equations by factorisation:
\((x3)\ (x4)=\dfrac{34}{(33)^2} \)
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Solve the following quadratic equations by completing the squares: \(4x^2+4\sqrt3x+3=0 \)
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Dividing each term by 4 we get
Solve the following quadratic equations by completing the squares: \(\sqrt3x^2+10x+7\sqrt3=0 \)
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Solve the following quadratic equations by completing the squares: \(a^2x^2 – 3abx + 2b^2 = 0 \)
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Solve the following quadratic equations by completing the squares: \(y^2+\dfrac{1}{2}y1=0 \)
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Solve the following equations by using the formula: \(x^2 – 2ax + 3x – 6a = 0 \)
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Solve the following equations by using the formula: \(a^2b^2x^2 + b^2x – a^2x – 1 = 0 \)
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Solve the following equations by using the formula: \(a^2 – b^2 = 4ax – 4x^2 \)
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Solve the following equations by using the formula: \((a – b)x^2 + (b – c)x + (c – a) = 0 \)
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Solve the following equations by using the formula: \((m^2+n^2)x^2+(m+n)x+\dfrac{1}{2}=0 \)
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Find the values of \(k \) for which the roots are real and equal in each of the following equations :
\((k + 1)x^2 + 2(k + 3)x + (k + 8) = 0 \)
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Find the values of \(k \) for which the roots are real and equal in each of the following equations :
\(5x^2 – 4x + 2 + k(4x^2 – 2x – 1) = 0 \)
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Find the values of \(k \) for which the roots are real and equal in each of the following equations :
\((4 – k)x^2 + (2k + 4)x + (8k + 1) = 0 \)
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For what value of \((4 – k)x^2 + (2k + 4)x + (8k + 1) = 0, \) is a perfect square.
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If the roots of the equations \(ax^2+2bx+c=0 \) and \(bx^22\sqrt{ac}x+b=0 \) are simultaneously real, then prove that \(b^2=ac \).
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and
and …(3)
…(4)
From eqn (3) and (4)
(Proved)
For what value(s) of \(k \), the equation \(x^22kxk=0 \) will have equal roots?
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Solve the equation : \(10ax^2+15ax6x9=0,\ a\ne0 \)
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Find the value of k for which the equation : kx (x – 2) + 6 = 0 has equal roots.
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For what value of \(k \) the equation \(4x^2 – 2\ (k +1) x + (k + 1) = 0 \) has real and equal roots ?
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Find the roots of the following quadratic equation \((x+3)\ (x+1)=3\ \left(x\dfrac{1}{3}\right) \)
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Find the value of \(k \) such that the quadratic equation \(x^22\ kx+(7k12)=0 \) has real and equal roots.
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Find the nature of roots of the quadratic equation \(\sqrt2x^2\dfrac{3}{\sqrt2}x+\dfrac{1}{\sqrt2}=0 \)
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The roots are equal and distinct in nature